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Topics Table of ContentsClick on a topic to go to that sectionDetermining if a Triangle is Possible
Geometric Constructions: The Basics
Videos Table of ContentsVideos Using Geometer's Sketchpad
Video: Constructing Circles
Video: Constructing Isosceles Triangles
Video: Constructing Equilateral Triangles
Video: Congruent Triangles
Glossary
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Topics Table of ContentsClick on a topic to go to that sectionDetermining if a Triangle is Possible
Geometric Constructions: The Basics
Videos Table of ContentsVideos Using Geometer's Sketchpad
Video: Constructing Circles
Video: Constructing Isosceles Triangles
Video: Constructing Equilateral Triangles
Video: Congruent Triangles
Glossary
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Constructions are shown within the lesson using paper and pencil. There are also video links showing how to do the constructions using Geometer's Sketchpad. A free version of this program is available at:
http://info.mheducation.com/sketchpad.trial.html
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How many different acute triangles can you draw?
How many different right scalene triangles can you draw?
Recall that triangles can be classified according to their side lengths and the measure of their angles.
Sides:Scalene - no sides are congruentIsosceles - two sides are congruentEquilateral - all three sides are congruent
Angles:Acute - all three angles are acuteRight - contains one right angleObtuse - contains one obtuse angle
Drawing Triangles
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There is another property that applies to triangles:
Click the lab below to determine what the Triangle Inequality Property is about.
Triangle Inequality Lab
Triangle Inequality Property
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Triangle Inequality: The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
What does this mean?
If you take the three sides of a triangle and add them in pairs, the sum is greater than (not equal to) the third side.
If that is not true, then it is not possible to construct a triangle with the given slide lengths.
Triangle Inequality Property
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ExampleDetermine if sides of length 5 cm, 8 cm and 12 cm can form a triangle?
Test all three pairs to see if the sum is greater:5 + 8 > 12 5 + 12 > 8 8 + 12 > 5 13 > 12 17 > 8 20 > 5
Yes, it is possible to construct a triangle with sides of lengths 5 cm, 8 cm and 12 cm.
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ExampleDetermine if sides of length 3 ft, 4 ft and 9 ft can form a triangle?
Test all three pairs to see if the sum is greater:3 + 4 > 9 3 + 9 > 4 4 + 9 > 3 7 > 9 12 > 4 13 > 3
No, it is not possible to construct a triangle with sides of lengths 3 ft, 4 ft and 9 ft.
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Try TheseDetermine if triangles can be formed with the following side lengths:
1. 4 cm, 7 cm, 10 cm 2. 24 mm, 20 mm, 30 mm
4 + 7 > 10 24 + 20 > 30 4 + 10 > 7 24 + 30 > 20 7 + 10 > 4 20 + 30 > 24 YES YES
3. 7 ft, 9 ft, 16 ft 4. 9 in, 13 in, 24 in
7 + 9 = 16 9 + 13 < 24 7 + 16 > 9 9 + 24 > 13 16 + 9 > 7 13 + 24 > 9 NO NO
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1 Determine if sides of length 5 mm, 14 mm and 19 mm can form a triangle. Be prepared to show your work!
YesNo
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1 Determine if sides of length 5 mm, 14 mm and 19 mm can form a triangle. Be prepared to show your work!
YesNo
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No
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2 Determine if sides of length 6 in, 9 in and 14 in can form a triangle. Be prepared to show your work!
YesNo
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2 Determine if sides of length 6 in, 9 in and 14 in can form a triangle. Be prepared to show your work!
YesNo
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Yes
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3 Determine if sides of length 5 yd, 13 yd and 21 yd can form a triangle. Be prepared to show your work!
YesNo
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3 Determine if sides of length 5 yd, 13 yd and 21 yd can form a triangle. Be prepared to show your work!
YesNo
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No
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4 Determine if sides of length 3 ft, 8 ft and 15 ft can form a triangle. Be prepared to show your work!
YesNo
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4 Determine if sides of length 3 ft, 8 ft and 15 ft can form a triangle. Be prepared to show your work!
YesNo
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No
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5 Determine if sides of length 5 in, 5 in and 9 in can form a triangle. Be prepared to show your work!
YesNo
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5 Determine if sides of length 5 in, 5 in and 9 in can form a triangle. Be prepared to show your work!
YesNo
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Yes
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6 A triangle could have which of the following sets of angles?
A 40º, 90º, 105º
B 35º, 89º, 56º
C 75º, 90º, 15º
D 30º, 65º, 95º
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6 A triangle could have which of the following sets of angles?
A 40º, 90º, 105º
B 35º, 89º, 56º
C 75º, 90º, 15º
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B
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7 A triangle could have which of the following sets of angles?
A 37º, 63º, 80º
B 90º, 104º, 76º
C 23º, 47º, 50º
D 80º, 90º, 10º
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7 A triangle could have which of the following sets of angles?
A 37º, 63º, 80º
B 90º, 104º, 76º
C 23º, 47º, 50º
D 80º, 90º, 10º
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A, D
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ExamplePredict the length of the third side of a triangle with sides of length 12 ft and 16 ft.
Side 1 = 12 ftSide 2 = 16 ft
The 3rd side must be less than: 12 + 16 > 3rd side 28 ft > 3rd side
The 3rd side must be greater than: 12 + 3rd side > 16 3rd side > 4
The 3rd side must be greater than 4 ft and less than 28 ft.
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ExamplePredict the length of the third side of a triangle with sides of length 9 cm and 15 cm.
Side 1 = 9 cmSide 2 = 15 cm
The 3rd side must be less than: 9 + 15 > 3rd side 24 cm > 3rd side
The 3rd side must be greater than: 9 + 3rd side > 15 3rd side > 6
The 3rd side must be greater than 6 cm and less than 24 cm.
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Try ThesePredict the length of the third side of a triangle whose known sides are lengths:
1. 13 mm, 20 mm 2. 7 in, 19 in
13 + 20 > Side 3 7 + 19 > Side 3 33 > Side 3 26 > Side 3
13 + Side 3 > 20 7 + Side 3 > 19 Side 3 > 7 Side 3 > 12
7 < side 3 < 33 12 < side 3 < 26
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Try ThesePredict the length of the third side of a triangle whose known sides are lengths:
3. 4 ft, 11 ft 4. 23 cm, 34 cm
4 + 11 > Side 3 23 + 34 > Side 3 15 > Side 3 57 > Side 3
4 + Side 3 > 11 23 + Side 3 > 34 Side 3 > 7 Side 3 > 11
7 < side 3 < 15 11 < side 3 < 57
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8 Predict the lower limit of the length of the third side of a triangle whose known sides are lengths 6 m and 12 m.
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8 Predict the lower limit of the length of the third side of a triangle whose known sides are lengths 6 m and 12 m.
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6 m
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9 Predict the upper limit of the length of the third side of a triangle whose known sides are lengths 6 m and 12 m.
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9 Predict the upper limit of the length of the third side of a triangle whose known sides are lengths 6 m and 12 m.
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18 m
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10 Predict the lower limit of the length of the third side of a triangle whose known sides are lengths 9 in and 17 in.
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10 Predict the lower limit of the length of the third side of a triangle whose known sides are lengths 9 in and 17 in.
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8 in
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11 Predict the upper limit of the length of the third side of a triangle whose known sides are lengths 9 in and 17 in.
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11 Predict the upper limit of the length of the third side of a triangle whose known sides are lengths 9 in and 17 in.
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26 in
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12 Predict the lower limit of the length of the third side of a triangle whose known sides are lengths 15 ft and 43 ft.
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12 Predict the lower limit of the length of the third side of a triangle whose known sides are lengths 15 ft and 43 ft.
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28 ft
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13 Predict the upper limit of the length of the third side of a triangle whose known sides are lengths 15 ft and 43 ft.
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13 Predict the upper limit of the length of the third side of a triangle whose known sides are lengths 15 ft and 43 ft.
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58 ft
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In Geometry, we can draw just about every figure with various tools. The tools that we will be using are given on this slide & the next slide:
1) Compass : creating circles & arcs
Geometric Tools
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In Geometry, we can draw just about every figure with various tools. The tools that we will be using are given on this slide & the next slide:
1) Compass : creating circles & arcs
Geometric Tools
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Show the students how the compass works by rotating it to form a circle.
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2) Ruler : measure segmentsD E DE = 6 cm
B C
A
m∠ABC = 65°
3) Protractor: measure angles
Geometric Tools
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Example Draw a circle that has a radius of 6 cm.
Step #1: Draw a segment w/ the ruler that measures 6 cm.
Step #2: Line up your compass so that it's center tip lies on one endpoint & the pencil tip lies on the other endpoint.
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Step #3: Keeping the distance between the center & endpoint the same, draw your circle.
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Step #3: Keeping the distance between the center & endpoint the same, draw your circle.
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the compass firmly in place & rotate the paper beneath the compass.
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Try this!1) Construct a Circle that has a radius of 3 cm using a ruler & a compass.
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Try this!1) Construct a Circle that has a radius of 3 cm using a ruler & a compass.
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Circulate around the room to make sure that the students are constructing the circles correctly. Figures should resemble the figure below.
3 cm
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Try this!2) Construct a Circle that has a radius of 8 cm using a ruler & a compass.
Video: Constructing Circles
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Try this!2) Construct a Circle that has a radius of 8 cm using a ruler & a compass.
Video: Constructing Circles
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Circulate around the room to make sure that the students are constructing the circles correctly. Figures should resemble the figure below.
8 cm
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A B
Example Use a ruler & a compass to draw an isosceles triangle w/ the following conditions: 1. at least one of the sides is 7 cm 2. at least one of the sides is 3 cm
Step #1: Look at your conditions. Both of them say "at least one" which means that one side, or more sides could meet the conditions. Plus, since the triangle is isosceles, we know that two of the sides must be equal. So pick which number you want to occur for 2 of your sides.
I'll select the 7 cm to occur twice.
Step #2: Draw one of your 7 cm segments.
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A B
Step #3: With your compass, draw a circle or semicircle (whichever you prefer).
Any segment that I connect from this arc to the center will have the same radius length of 7 cm.
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Step #4: With your ruler, find the segment that can be drawn from B to another point on the semicircle so that the ruler measures 3 cm. Make a point at this location.
A B
Step #5: Connect this point with points A & B to form your triangle.
A B
C
Note: AC = AB = 7 cm, since they are both radii of the circle.
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Think about this...
Can we make a triangle if we use the 3 cm twice and the 7 cm once?
Discuss this problem in your groups for a few minutes.
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Think about this...
Can we make a triangle if we use the 3 cm twice and the 7 cm once?
Discuss this problem in your groups for a few minutes.
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No:3 + 3 > 76 > 7
?
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Try this!3) Use a ruler & a compass to draw an isosceles triangle w/ the following conditions: 1. at least one of the sides is 8 cm 2. at least one of the sides is 6 cm
Video: Constructing Isosceles Triangles
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Try this!3) Use a ruler & a compass to draw an isosceles triangle w/ the following conditions: 1. at least one of the sides is 8 cm 2. at least one of the sides is 6 cm
Video: Constructing Isosceles Triangles
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Circulate around the room to make sure that the students are constructing the triangle correctly. Note: 2 answers (could ask them to find both for extra practice): 8 cm, 8 cm & 6 cm & 6 cm, 6 cm & 8 cm
8 cm
8 cm
6 cm
6 cm
6 cm
8 cm
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Example Use a ruler & a protractor to draw an equilateral triangle that has a side length of 5.5 cm.
Step #1: Reread your question. We have an equilateral triangle, which has all of the sides measuring 5.5 cm & all of its angles measure to be 60°.
Step #2: Draw one of your 5.5 cm segments.
C D
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C D
Step #2: With your protractor, place its center at either point C or point D (doesn't matter), measure 60° & draw a line connecting the 60° measurement with your center (mine is C).
Step #3: Repeat step #2 with your other endpoint as the center.
C D
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C D
Step #4: Draw a point where the 2 lines intersect & erase any additional lines.
Note: You can verify that all of the edges are equal by measuring all of them.
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Try this!4) Construct an equilateral triangle that has a side length of 4.2 cm using a protractor and a ruler.
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Try this!4) Construct an equilateral triangle that has a side length of 4.2 cm using a protractor and a ruler.
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Circulate around the room to make sure that the students are constructing the triangles correctly. Diagrams should resemble the figure below.
4.2 cm
4.2 cm4.2 cm
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Try this!5) Construct an equilateral triangle that has a side length of 6.7 cm using a protractor and a ruler.
Video: Constructing Equilateral Triangles
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Try this!5) Construct an equilateral triangle that has a side length of 6.7 cm using a protractor and a ruler.
Video: Constructing Equilateral Triangles
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Circulate around the room to make sure that the students are constructing the triangles correctly. Diagrams should resemble the figure below.
6.7 cm
6.7 cm6.7 cm
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Example Use a ruler & a protractor to draw ∆ FGH that satisfies the following conditions. 1. m∠F = 30° 2. m∠G = 70° 3. FG = 8 cm
Step #1: Reread your question. We know two angle measurements and the length of 1 side. Let's start with the side, since at least 1 segment is required to use a protractor.
Step #2: Draw your 8 cm segment. Label one endpoint with F & the other with G
F G
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F G
Step #2: With your protractor, place its center at point F. Measure the 30° required for m ∠F. Draw a line through the 30° angle
F G
Step #3: With your protractor, place its center at point G. Measure the 70° required for m ∠G. Draw a line through the 70° angle
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Step #4: Draw point H where the 2 lines intersect & erase any additional lines.
F G
H
30° 70°8 cm
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Try this!6) Construct ∆JKL with the given conditions using a protractor and a ruler. 1. m∠K = 105° 2. m∠L = 25° 3. KL = 9 cm
Video: Constructing Triangles
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Try this!6) Construct ∆JKL with the given conditions using a protractor and a ruler. 1. m∠K = 105° 2. m∠L = 25° 3. KL = 9 cm
Video: Constructing Triangles
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Circulate around the room to make sure that the students are constructing the triangles correctly. Their diagrams should resemble the figure below.
L K
J
9 cm105°25°
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Glossary
Return toTable ofContents
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in the presentation. The text box the word is in is then linked to the page at the end of the presentation with the word defined on it.
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Back to
Instruction
Compass
An instrument with 2 arms, one sharp & one with a pencil that can be used to
draw circles and arcs.
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Back to
Instruction
Protractor
An instrument used to measure angles. They are usually half of a circle (180°). To measure an angle, line up the protractor so that the vertex of the angle aligns with the small hole, or dot, near the bottom of the protractor and one of the angle
edges lies on the straight line at the bottom.
30°
90°
135°
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Back to
Instruction
Ruler
An instrument used to measure the lengths of segments.
A
B
C
D
E
F
AB = 3.5 cm
CD = 5 cm
EF = 6.5 cm
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Back to
Instruction
5
103
510
3
5 + 3 > 10
Triangle Inequality
The sum of the lengths of any two sides of a triangle is greater than the length
of the third side.
6
104
610
4
6 + 4 > 10
7
105
7
10
5
7 + 5 > 10
