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Geometry EOC Review Packet

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The Review Packet Directions

1) Number each page of the packet and use your numbering scheme to make a table

of contents (below).

2) Next, use your numbering scheme to make an index for all of the topics. Attach

your index to the last page of the packet. (Hint there will be over 100 topics)

Table of Contents

Cover Page ……………………………………………………………………… 1

Part 1: Points, Lines, Planes, and Angles

The Distance

Formula

If you are given two points and , you can use the distance formula to find the distance between the two points. It is based on the Pythagorean Theorem.

In the example above, let point 1 be (-2,3) and point 2 be (4,-3). Find the distance between these two points using the distance formula.

The

Midpoint

Formula

The point exactly in the middle of a segment, halfway from either endpoint. If you are given two points and , you can use the midpoint formula to find the distance between the two points. Example: Find the midpoint for the segment with endpoints at: (-2,3) and (4,-3)

The

Midpoint

Formula (continued)

Sometimes, you will be given the midpoint of a segment and you will have to find one of the endpoints of the segment. You can still use the formula. Example: The midpoint of a segment is at (3,-5). One of the endpoints is at (2,-2). Find the other endpoint. The midpoint is (3,-5). So set (3,-5) equal to the formula. And say the given point (2,-2) is

So the other endpoint must be (4,-8)

Angle Pairs

formed by a

Transversal

Given: line l is parallel to line m and line t is a transversal

Angle Pairs are divided up into congruent pairs and supplementary pairs like this:

Congruent Supplementary

Alternate Interior Angles

2 and 6

7 and 3

Same Side Interior Angles

2 and 3

7 and 6 Alternate Exterior Angles

1 and 5

8 and 4

Same Side Exterior Angles

1 and 4

8 and 5

Corresponding Angles

1 and 3 or 8 and 6

2 and 4 or 7 and 5

Linear Pairs

1 and 2 or 1 and 8

8 and 7 or 7 and2

Now, break this into 2 separate

equations and solve for x2 and y2

Part 2: Polygons

Polygon

A two-dimensional closed shape with only straight lines (no curves). Polygons cannot cross themselves anywhere (like

Figure 2 for Not a Polygon).

Polygon Not a Polygon

Convex vs

Concave

There are two types of polygons. Convex polygons and concave polygons. Convex polygons are polygons whose interior angles are each less than 180°. Concave polygons, on the other hand, have an angle formed within the shape.

Convex Concave

Convex vs

Concave (continued)

Convex Concave

Regular

Polygons

Regular polygons have all sides and all angles the same. A regular quadrilateral would be a square. A regular triangle would be equilateral. Every other type of polygon is considered irregular.

Sum of the

Interior

Angles

The sum of the interior angles of a polygon are given as follows:

Start with a triangle: 180. Add another 180 for each additional side the shape has.

Shape Number of Sides Sum of Angles

Triangle 3 180

Quadrilateral 4 180 + 180 = 360 Pentagon 5 540

Hexagon 6 720

Heptagon 7 900 n-gon N (n-2)180

Example: a shape with 10 sides has 8(180)=1440

Sum of the

Exterior

Angles

The Exterior Angles of a polygon, are the ones found by extending each of the sides. Then the exterior angles are supplementary to the interior angles.

No matter the shape, the sum of the exterior angles is always

360.

Congruent

Polygons

Congruent polygons have all the same angles and sides. Imagine you can lay one right on top of the other.

Similar

Polygons

Two polygons that are the same shape but have different sizes. The corresponding angles will be congruent.

~ You can set up a ratio. The smaller side over the bigger side.

Scale

Factor

Scale factor is the ratio of the smaller shape to the larger

shape. It can be written as a ratio 1:3 or as a fraction ratio

That means, the larger shape’s sides are 3 times as big as the smaller shape’s sides. The scale factor for area is squared for both numerator and denominator (1:9) and cubed for volume.

Perimeter The length around the outside of a figure. Perimeter is found by adding the length of each external side of a figure.

Area Area is the amount of space inside a figure. Most common figures have a formula used to find their area.

Area Formulas

Rectangle

Triangle

Parallelogram (Note: the height is not the

length of the outer sides, it is the altitude of the parallelogram)

Rhombus

(where d1 and d2 are the

diagonals of the rhombus)

Trapezoid

Regular Polygon

(apothem is the perpendicular distance from the side of a regular polygon to its center)

Part 3: Quadrilaterals

Quadrilateral A quadrilateral is a polygon with four sides.

Parallelogram

A type of quadrilateral that has both sets of opposite sides parallel

Properties:

m Both pairs of opposite sides are congruent m Consecutive angles are supplementary m Opposite angles are congruent m The diagonals bisect each other

Rectangle

A type of parallelogram that has all right angles.

Properties:

m Both pairs of opposite sides are congruent m Opposite sides are parallel m All angles are right m Diagonals bisect each other m Diagonals are congruent

Rhombus

A type of parallelogram that has all congruent sides.

Properties:

m All sides are congruent m Opposite sides are parallel m Opposite angles are congruent m Consecutive angles are supplementary m Diagonals are perpendicular bisectors m Diagonals bisect the angles

Square

A type of parallelogram that is both a rectangle and a square. It has four right angles and all congruent sides.

Properties:

m All sides are congruent m Opposite sides are parallel m All angles are right m Consecutive angles are supplementary m Diagonals are perpendicular bisectors m Diagonals bisect the angles

Trapezoid

A trapezoid is a quadrilateral with only one set of parallel sides. These sides are called the bases.

Properties:

m Only one pair of sides is parallel m Consecutive angles along the non-parallel

sides are same-side interior angles and therefore supplementary

Isosceles

Trapezoid

An isosceles trapezoid is a quadrilateral with only one set of parallel sides called the bases. But the non-base sides are congruent.

Properties:

m Only one pair of sides is parallel m Non-parallel sides are congruent m Base angles are congruent m Diagonals are congruent m Opposite angles are supplementary

Kite

A kite is a quadrilateral with no sets of parallel lines. There are two sets of adjacent sides congruent. The angles where the non-adjacent sides meet are congruent and the diagonals meet at a right angle in the middle.

Note: Opposite sides are neither parallel nor

congruent.

Part 4-A: Triangles

Classifying

Triangles

Triangles are classified in two ways: by the lengths of their sides or by their angle measures.

Classification by Side Length

Equilateral All 3 sides are congruent

Isosceles 2 sides are congruent

Scalene No sides are congruent

Classification by Angle Measure

Acute Only acute angles in the triangle

Right One right angle

Obtuse One obtuse angle

Equiangular All 3 angles congruent

Altitude of a

Triangle

A perpendicular segment from a vertex to its opposite side.

Every triangle has three altitudes. The altitude is also called the height and the altitude is what is needed for

the area of a triangle. A =

Median of a

Triangle

A line segment that joins a vertex of a triangle to the midpoint of its opposite side. (Every triangle has 3)

Angle

Bisector

A segment that passes through the vertex of a triangle to the opposite side and cuts the angle in half.

Every triangle has

three angle

bisectors

Perpendicular

bisector

A line that passes through the midpoint of a side and is perpendicular to that side. Every triangle has three perpendicular bisectors.

Points of

Concurrency

A concurrent point is where three or more lines (or segments) intersect. Every triangle has three of each of the types of segments listed above, that is: three altitudes, three medians, three angle bisectors, and three perpendicular bisectors. If you draw all three medians at once, the place where they meet is called a concurrent point. There are four types of concurrent points in triangles:

Orthocenter The point of concurrency for the 3 altitudes of a triangle. It can occur inside, on, or outside the triangle itself.

Centroid The point of concurrency for the 3 medians of a triangle. The centroid is always inside the triangle. Important: A median can be broken

into thirds where

of the segment

lies between the vertex and the

centroid and

of the segment lies

between the side and the centroid. Incenter The point of concurrency for the 3

angle bisectors of a triangle. The incenter is always inside the triangle.

Circumcenter The point of concurrency for the 3 perpendicular bisectors of a triangle. The circumcenter can occur inside, or, or outside the triangle itself.

Part 4-B: Congruent Triangles

Congruent

Triangles

Triangles are congruent when all corresponding sides and corresponding angles are congruent. They must be the exact same shape and size.

Corresponding Sides: Corresponding Angles:

Altogether a triangle has six measurements: three sides and three angles. For some pairs of triangles though, you only need three of those measurements to prove the triangles are congruent.

sSS The Side-Side-Side Postulate says that if all three sets of corresponding sides are equal, then the triangles must be congruent.

and

sAS The Side-Angle-Side Postulate says that if only two pairs of corresponding sides are congruent and their included angles are equal, then the triangles are congruent.

or

ASA The Angle-Side-Angle Theorem says that if two pairs of corresponding angles and their included side are congruent, then the triangles must be congruent.

or

AND

AAS The Angle-Angle-Side Theorem says that if two pairs of corresponding angles and a pair of non-included sides are congruent, then the triangles must be congruent.

Proving

Triangles

are

Congruent

Prove

Statements Reasons

Given

are alternate interior angles

They are on opposite sides of the transversal

Alternate interior angles are congruent

are alternate interior angles

They are on opposite sides of the transversal

Alternate interior angles are congruent

Reflexive Property

ASA Theorem

In words: Because and are parallel, we can identify some angle pairs. On opposite sides of the transversal there are two sets of alternate interior angles. That makes two pairs of congruent angles, all we need now to prove the triangles congruent is a pair of congruent sides. is part of both triangles. That means the triangles have a congruent side (it happens to be the same exact thing but part of both triangles). Since is the side between the two angles, we must conclude that the triangles are congruent by ASA Theorem.

HL The Hypotenuse-Leg Theorem is only for right triangles. If the hypotenuses of two right triangles are congruent and one pair of corresponding legs are congruent, then the triangles must be congruent.

Part 4-c: Similar Triangles

Similar

Triangles

Triangles are similar when all corresponding angles are congruent and when the corresponding sides are proportional—that is, they can be written as the exact same ratio. Remember, every triangle has three sides, so to determine which sides correspond to one another it helps to make a table.

aA ~

The Angle-Angle Similarity Postulate says that if two pairs of corresponding angles are equal then the triangles are similar.

sSS ~ The Side-Side-Side Similarity Postulate says that if all three pairs of corresponding sides are the same ratio, then the triangles must be similar.

Smallest Sides

Medium Sides

Largest Sides

Smallest Sides

Medium Sides

Largest Sides

4:6 8:12 10:15

All of the ratios are the same, so the

sides are all proportional, meaning

the triangles are similar by SSS

sAS ~ The Side-Angle-Side Similarity Postulate says that if there is one pair of corresponding angles in a pair of triangles, and the two sides that form those angles have the same ratios, then the triangles must be similar.

Measurements

In similar

Triangles

We already know that the corresponding sides of similar triangles have the same ratio. We also know their corresponding sides are congruent. If the ratio of the

corresponding side lengths is

, then the areas must be

that ratio squared

Example: The scale factor (another name for the ratio between sides of similar triangles) is 1:5 meaning, the legs of the second triangle are 5 times bigger than the first.

The fraction for this ratio is

.

The area of the second triangle then, is given by the

ratio

meaning it is 25 times bigger than the first.

So if the area of the smaller triangle is 10 feet2, then the area of the bigger triangle is 250ft2

Smallest Sides

Medium Sides

Largest Sides

6:15 8:20

The ratios of two sides are the same and the angle between the two

sides is equal to itself, therefore the triangles are similar by SAS~

Part 4-D: Triangle Theorems

Proportional

segments

Theorem 1

If a line is parallel to one side of a triangle and intersects the other two sides, then it divides the sides proportionally.

5x=40

x=8

Proportional

segments

Theorem 2

If a ray bisects one angle of a triangle, then it divides the opposite side into segments that are proportional to the two sides that form the bisected angle.

660a=10

a=66

10

8.8

7.5

x

a

Triangle

Inequality

Theorem

The sum of the lengths of any two sides of a triangle must be greater than the third side.

Example: The length of two sides of a triangle are 5 and 8 inches. Which of the following is a possible length for the third side?

a) 10 inches b) 15 inches c) 3 inches

Answer: Call the third side x. We have the following inequalities: 5 + x > 8 (Which means x has to be greater than 3) 8 + x > 5 (This is true for all positive numbers) 5 + 8 > x (So x must be less than 13) The only answer that meets this criteria is a) 10 in.

Side Angle

Inequality

Theorem

In a triangle, the largest angle is across form the longest side and the smallest side is across from the smallest angle.

Exterior Angle

Inequality

Theorem

The measure of an exterior angle of a triangle (the angle that is formed by extending a side) is greater than the measure of either remote interior angle.

Hinge Theorem The Side Angle Inequality Theorem applies in separate triangles. If two triangles have two sides exactly the same, but the angle between those sides is different, then the bigger angle is still across from the bigger side.

is the longest side

because it’s across from

the biggest angle. is

the shortest side.

x > a

AND

x > b

3 + 4 > 6

3 + 6 > 4

4 + 6 > 3

Since A is bigger

than D and

everything else is

the same, then

Part 5-A: Right Triangles

The

Pythagorean

Theorem

C is the length of the hypotenuse of the right triangle. Example: Find the length of the missing side:

Pythagorean

Theorem

Word

Problems

Anna lives at point A. She walks 2 kilometers North and 5 kilometers East to visit her friend Celia who lives at point C. To the nearest tenth many kilometers would Anna have saved if she had walked straight to Celia’s house?

Anna walked 7 kilometers

altogether. If she had walked

straight from her house to

Celia’s, she would have walked

the hypotenuse of a right

triangle, so use the

Pythagorean Theorem to solve:

Altogether, Anna would have saved 7 – 5.4 = 1.6 kilometers.

Acute, right, or obtuse

triangles

Use the same formula from the Pythagorean Theorem, labeling the sides a, b, and c where a is the smallest side, b is the medium side, and most importantly, c is always the longest side of the triangle. If c2 is bigger than a2 + b2, then the triangle is obtuse. If c2 is smaller than a2 + b2 then the triangle is acute. If c2 is equal to a2 +b2 then the triangle is right.

Obtuse Acute Right

c2 > a2 + b2 c2 < a2 + b2 c2 = a2 + b2

Example: Is a triangle with sides 4, 5, and 6 acute, obtuse, or right? 62 ___ 42 + 52

36 ___ 16 + 25 36 ___ 41 36 _<_ 41 So the triangle is acute.

Altitude to

the

Hypotenuse

Drawing in an altitude of a right triangle from the hypotenuse to the right angle creates three similar right triangles.

Example: Solve for x in the triangle below.

m We can write a ratio with the smallest side of the small triangle

over the smallest side of the large triangle.

m We can write a ratio with the largest side of the small triangle

over the largest side of the large triangle.

m Set the proportions equal to each other and solve for x.

Smallest Sides Medium Sides Largest Sides

Smallest Sides Medium Sides Largest Sides

Cross multiply to get:

so

Part 5-B: Special Right Triangles

Special Right

Triangles

There are two types of special right triangles. They are named for the number of degrees in each of the three angles.

m 30-60-90 Triangle m 45-45-90 Triangle

30-60-90 Triangle

A 30-60-90 triangle is actually formed by dividing an equilateral triangle in half.

In the example at the right, an equilateral triangle with each side length 10 has been split down the

middle into two 30-60-90 triangles. The shortest side

of the new triangle is half the length of the hypotenuse.

Using the formula in the box above, we can solve for the length of the missing side of this triangle.

Say x = 5

The third side is simply x or 5

Example: Use the formulas for a 30-60-90 triangle to find the missing sides of this triangle:

In this example, the hypotenuse is 16. So using the formulas in the box above we know 16 = 2x. That means x = 8.

Therefore, the shortest side is 8. Also, we know the medium side is x or 8

45-45-90 Triangle

A 45-45-90 triangle is actually just an isosceles right triangle.

The two legs of this right triangle are congruent.

The hypotenuse of a 45-45-90 triangle will always be the length of a

leg multiplied by That means, the length of the hypotenuse in the

triangle above is Example: Use the formula for a 45-45-90 triangle to solve for the missing sides in the triangle below.

The length of one of the legs is 8. Since a 45-45-90 triangle is isosceles, that means the other leg must also be 8. Using the formula for a 45-45-90

triangle, we know the hypotenuse must be

Rationalizing

the

Denominator

Sometimes, when we use the special right triangle formulas, to solve for x, we will end up dividing by

either or . When that happens, we will end up

with a fraction that has either or in the denominator. This is bad math grammar. We will need to multiply by a factor of 1 to write the answer properly.

Example: Solve for x in the triangle below.

The hypotenuse of this 45-45-90 triangle is 6. That means:

This comes from the

formula.

Divide both sides by

cancels on the right.

Here we cannot leave in the denominator. So

multiply by

Part 5-C: Trigonometry

Sine, Cosine, Tangent

The trigonometric ratios (sine, cosine, and tangent) are a way of relating the sides of a right triangle to its angles.

When solving a problem of this sort, you will either have to solve for the length of a missing side or for a missing angle. Note: when solving for a missing angle, you will need to use what’s called the inverse sine, cosine, or tangent. ( )

It helps sometimes to list all of the given information ahead of time.

Angle Side Opposite

the Angle Side Adjacent to the Angle

Hypotenuse

Sine

The sine of an angle is the ratio of the length of the leg opposite that angle divided by the length of the hypotenuse.

sin45 =

Cross multiply to solve for x. We end up with

An O A H

45 x -- 8

An O A H

x 9 -- 12

Take the inverse

sine (sin-1) of both

sides of the

equation to get x

by itself.

Cosine

The cosine of an angle is the ratio of the length of the leg adjacent to the angle (the leg that is touching the angle) divided by the length of the hypotenuse.

Cross multiply to solve for x.

We end up with

We can use inverse cosine to solve for an angle measure.

Tangent

The tangent of an angle is the ratio of the length of the leg opposite the angle divided by the length of the leg adjacent to the angle.

Cross multiply to solve for x. We end up with We can use the inverse tangent to solve for an angle measure.

An O A H

50 -- 10 x

An O A H

x -- 7 14

An O A H

35 x 10 --

An O A H

x 13 6 --

Angle of

Elevation

Problems

Word problems for sine, cosine, and tangent that involve looking up at something. Example: A man who’s eye level is 6 feet above the ground looks up at an eagle in a tree. The tree is 20 feet away from the man and the angle at which he is looking

is 30. Find how high the eagle is from the ground.

Cross multiply to solve for x. We get feet. The final answer would be 11.5+6=17.5 feet.

Angle of

Depression

Problems

Word problems for sine, cosine, and tangent that involve looking down at something. Note: the angle of elevation (in the box above) is generally inside the right triangle formed on the figure. The angle of depression is usually outside the triangle.

Ex: A man looks down at a 50 angle and sees a shark.

We can draw a right triangle on this

figure with 30 for the angle of

elevation. Note: The bottom leg of

the triangle must be 20 feet since

the distance below it is also 20 feet.

An O A H

30 x 20 --

Part 6-A: Polyhedrons and Solids

Polyhedron Polyhedron is an enclosed solid figure with all flat surfaces. (Note: Cylinders, cones, and spheres are not polyhedrons because they involve circles). Polyhedrons are like three dimensional polygons—their faces are all made of polygons.

Types of

Solid

Figures

Description Net (What it looks like all opened up and laid

out flat)

A prism has a pair of

parallel polygon bases.

A pair of parallel circular bases.

One polygon base.

One circular base.

A set of points in space

equidistant from a center

None.

Base

Base

Euler’s

Formula

Euler’s Formula is a formula that relates the number of faces (F), vertices (V), and edges (E) of a polyhedron. Faces (F) are the flat surfaces of a polyhedron (the polygons). A cube has 6 faces that are all square. Vertices (V) the points formed by the intersection of three or more edges. A cube has 8 vertices. Edges (E) line segments formed by the intersection of two faces. A cube has 12 edges.

Euler’s Formula says that the number of vertices (V) plus the number of faces (F) minus the number of edges (E) equals 2.

V + F – E = 2

For a cube, we have: 8 + 10 – 12 = 2 Example: A certain polyhedron has 4 faces and 8 edges, how many vertices does it have? V + F – E = 2 V + 4 – 8 = 2 V – 4 = 2 V = 6 (There are 6 Vertices)

face

vertex

edge

Part 6-B: Surface Area and Volume

Surface Area The sum of the areas of all the faces including the base(s). Surface area is measured in square units. Imagine surface area is the amount of wrapping paper you would need to wrap the object.

Volume The amount of space inside of a solid. Volume is measured in cubic units. Imagine volume as how much water it would take to fill the item. Sometimes we measure volume in gallons (like how many gallons would we need to fill a certain fish tank), but here we will measure volume in cubic meters, centimeters, or sometimes even cubic feet.

Rectangular

Prism Surface Area:

Volume:

Cylinder

Surface Area:

Volume:

Right Square

Pyramid

Surface Area:

(Where P is

the perimeter of the base, B is

the area of the base, and l is

the slant height)

Volume:

(Where B is the area

of the base)

Cone

Surface Area: (Where l

is slant height)

Volume:

Sphere

Surface Area:

Volume:

Example Find the Volume and Surface Area of the square pyramid below:

m The height is 8 feet and the width of the square base is 12 feet.

m That means we can use the Pythagorean Theorem to find the slant height.

Surface Area Volume

m P (for Perimeter of Base)

is 12 + 12+ 12 + 12 = 48 m L (for slant height) is 10 m B (for Area of Base) is

m B (for Area of Base) is

m H (for height) is 8

Note: S.A. and V happen to be the same in this problem. This is only a coincidence. They are usually different.

Part 7: Conditional Statements

Conditional

Statement

A conditional statement is a statement (that may be true or false) that has an “if” and a “then” in it. Example: “If a shape is a square, then it is a type of rectangle.” We use the notation:

Truth Value The truth value is simply “True” or “False”. Every statement has a truth value that is either “True” or “False” Example: “If a shape is a square, then it is a type of rectangle.” The truth value of this statement is “True” Example: “If a shape is a rectangle, then it is a type of square.” The truth value of this statement is “False” because not all rectangles are necessarily squares.

Hypothesis The hypothesis of a conditional statement is the part following the “if”. We label this part P Example: “If a shape is a square, then it is a type of rectangle.” The hypothesis is “A shape is a square.”

Conclusion The conclusion of a conditional statement is the part following the “then”. We label this part Q Example: “If a shape is a square, then it is a type of rectangle.” The conclusion is “It is a type of rectangle.”

Converse The converse of a conditional statement is what you get when you reverse the order of the hypothesis and the conclusion. We use the notation: Conditional Statement: “If a shape is a square, then it is a type of rectangle.” Converse: “If a shape is a rectangle, then it is a type of square.”

Note: The converse is not always true when the conditional is true.

Inverse The inverse of a conditional statement is what you get when you add the word “not” to both the hypothesis and the conclusion. We use the notation because ~ represents the word “not”. Conditional Statement: “If a shape is a square, then it is a type of rectangle.” Inverse: “If a shape not is a square, then it is not a type of rectangle.”

Note: The inverse is not always true when the conditional is true. In fact, this inverse above has a truth value of “false” since there exist plenty of rectangles that are not squares.

Contrapositive The Contrapositive of a conditional statement is what you get when you both reverse the order of the hypothesis and conclusion and add the word “not” to the hypothesis and conclusion. We use the notation: Important: The Contrapositive of a statement is always true when the conditional is true and always false when the conditional is false. We say it is logically equivalent to the conditional statement. Conditional Statement: “If a shape is a square, then it is a type of rectangle.” Contrapositive: “If a shape is not a rectangle, then it is not a type of square.”

Note: The Contrapositive is always true when the conditional is true. In this example we know that all squares are types of rectangles, so if a shape is not a rectangle then it cannot possibly be a type of square.

Part 8: Circles

Radius Radius – Any segment in a circle whose endpoints lie a) somewhere on the circle itself and b) at the center of the circle

All radii (the plural of radius) are the same length so when you see the question “What is the radius of this circle?” it is talking about any radius in the circle.

Chord Chord – Any segment in a circle whose endpoints both lie on the circle itself.

Diameter Diameter – A chord that passes exactly through the center of a circle. It is double the radius.

Secant Secant – A chord that has been extended to a line. (A line that intersects a circle in two places)

Tangent Tangent – A line or segment that intersects a circle in exactly one place. Note: It is perpendicular to the radius.

r

Circumference

of a Circle

The Circumference is another name for the Perimeter of a Circle – that is, the distance around the outside of the circle. Circumference can be found using one of these formulas: or

The Circumference of the circle above is

Central

Angle

A central angle is an angle formed by two radii of a circle. The vertex of the angle is the circle’s center and the angle’s endpoints lie on the circle itself.

Note: In the box below we will have the definition for an arc, which is a part of a circle. The measure of an arc (in degrees) is always equal to the

measure of its central angle. So the measure of (read “arc AB”) is 60

Arc An arc is a part of a circle.

Semicircle An arc that is exactly half a circle. is a semicircle in the figure below. The measure

of is 180 Note: You need three points to name a semicircle in order to show which semicircle you mean exactly (top or

bottom in this case)

Minor Arc An arc that is smaller than a semicircle. is a minor arc whose measure

is 60 Note: You need only two points to

name a minor arc. With two points it is assumed that you mean the smallest

arc between those points.

Major arc An arc that is bigger than a semicircle. is

a minor arc whose measure is 120. is

a major arc whose measure is 360 – 120 = 240 Note: You need three points to name a major arc. Otherwise it is assumed you

mean the minor arc.

Arc Length Since an arc is only part of a circle, the length of an arc, is only part of the entire length around the circle (aka the Circumference). It is actually a fraction of the entire Circumference. The only question is: what fraction?

Example 1: In the circle below, represents a quarter of

the whole circle so we say the fraction is

. We must

multiply the circumference of the entire circle by

to find

what the length of . inches

Arc Length

Finally,

simplifies to just

in.

Example 2: In the circle below represents a fraction of the whole circle. But what fraction exactly? To figure that

out, we take the number of degrees in out of the total

number of degrees in the circle, that is:

Now, we just

need to simplify that fraction.

meters

Arc Length =

Finally,

cannot be simplified

further.

We can plug

into the calculator to get 14.65 meters

Area of a

Circle

The area of a circle is the amount of space inside a circle. It is given by the formula

Area is measure in square units.

Sector of a

Circle

A sector of a circle is a portion of the entire circle’s area. It is formed between two radii of a circle and their intercepted arc.

The sector formed at the left, by would be one quarter of the circle’s entire area. To find the area of that sector exactly,

mult. the area of the whole circle by

Area of the sector =

Since

cannot be simplified, just plug into the calculator

to get

Area of a

Sector

Just like with arc length, the area of a sector is found using a fraction where we need to figure out exactly which fraction. To do this, always take the number of degrees in the given section over the total number of degrees in a circle.

The sector formed at the left, by major

arc is 360 – 120 = 240 The fraction of the circle this sector

represents is

So, to find the area of that sector exactly, multiply the area

of the whole circle by the fraction

Area of the sector =

can be simplified to , then plug into the calculator

to get

6 cm

Segment of

a Circle

A segment of a circle is a portion of a sector of a circle. It is the part you get by drawing a line segment that connects the two endpoints of an arc.

The segment formed at the left, by

and is a part of the sector formed there. To find the area of that segment, you have to first find the area of the sector and then subtract the area of

Area of segment = Area of Sector – Area of

–

–

–

–

–

– The area of the sector then is

Inscribed

Angle

An inscribed angle is an angle whose vertex lies on the circle itself and whose endpoints also lay on the circle itself. It is an angle formed by two chords.

The measure of an inscribed angle is half the measure of

the intercepted arc. and are inscribed angles.

m Since is 40 we can conclude m is 20

m Since is 55 we can conclude m is 110

Angle

Inside a

Circle

An angle formed by 2 chords that cross inside a circle. The measure of the angle is the average of its two intercepted arcs. We can use the formula:

x =

where a and b are the

arc measures.

x =

Angle

outside a

circle

When two tangents, two secants, or a tangent and a secant meet outside a circle, we have a formula to figure out the measure of the angle they form. It is similar to taking the average where you add and divide by 2, except instead of adding, we subtract to find the difference between the bigger arc and the smaller arc, then divide by 2.

Formulas

for

Segments in

a Circle

Segments in a circle are formed by secants, tangents or chords. While before we were looking for angle measure, now we are looking for segment length.

Intersecting Chords

Example: Solve for x.

Intersecting Secants

Example: Solve for x.

Formulas

for

Segments in

a Circle (Continued)

Tangent and Secant Intersecting

Example: Solve for x.

Intersecting Tangents

Example: Solve for x.

Equation

of a Circle

The equation of a circle is the equation used to graph a circle on a coordinate plane. The important parts are the center of the circle (given by the coordinates: (h,k) and the radius of the circle, r. The equation looks like this:

Examples:

m The center of the above circle is (3,6) The radius is 10.

m The center of the above circle is (9,4) The radius is 5.

m The center of the above circle is (-2,5) The radius is 6.

Write a circle’s equation with center (-1,-3) and radius 2.