Finding Equations of Lines

If you know the slope and one point on a line you can use the point-slope form of a line to find the equation.

1 1y y m x x

If you know the intercepts of a line, you can use the intercept form of a line to find the equation.

1x y

a b intercept

intercept

a x

b y

Ex. 1 Find an equation of the line with x-intercept 8 and y-intercept 4.

Use the intercept form of a line. 8 and 4a b

18 4

x y

8( 1)8 4

x y 2 8x y

Ex. 2. Find the equation of the line through (1, 4) and (5, -2)

Find the slope m.4 2 6 3

1 5 4 2m

Use point-slope form of a line. 1 1y y m x x

34 1

2y x

3 34

2 2y x

3 11

2 2y x

2 3 11y x

3 2 11x y

Ex. 3. Find an equation of the line with y-intercept –3 and parallel to 2 5 8x y

Put original equation in slope-intercept form2 5 8x y 5 2 8y x

2 8

5 5y x

A parallel line will have the same slope.

Use the y-intercept given.

23

5y x

Put in general form.2

35

y x

5 2 15y x

2 5 15x y

A segment is a median of a triangle if and only if it extends from a vertex of a triangle to the midpoint of the opposite side.

Every triangle has three

A B

C

DE

F

The point of concurrency of the medians is called the centroid.

A segment is an angle bisector of a triangle if and only if it bisects an angle of the triangle and has 0ne endpoint on the opposite side.

A B

C

Every triangle has three.

The point of concurrency of the angle bisectors of a triangle is called the incenter.

An altitude is a segment from a vertex of a triangle perpendicular to the line containing the opposite side.

acuteA

B

C

DE

F

A

BCright

A

B

CD

E F

obtuse

Every triangle has three.

D

The point of concurrency of the altitudes is called the orthocenter.

A perpendicular bisector is perpendicular to and bisects the opposite side of the triangle.

A B

C

D

EF

The perpendicular bisectors intersect at a point that is equidistant from the vertices of the triangle.

The point of concurrency of the perpendicular bisectors of the sides of a triangle is called the circumcenter.

Ex. 4. Write an equation of the perpendicular bisector of the segment joining A(-2, 3) and B(4, -5)

Find the slope of the original segment.3 5 8 4

2 4 6 3m

The slope of the perpendicular bisector would be the negative reciprocal.3

4m

The perpendicular bisector intersects the segment at its midpoint.

2 4 3 5 of , 1, 1

2 2mp AB

Use point-slope form of a line. 31 1

4y x

3 31

4 4y x

3 7

4 4y x

4 3 7y x 3 4 7x y

Another way to do example 4 would be to use the distance formula.

Consider point P(x, y), if it is on the perpendicular bisector of AB, then AP = PB.

AP AB

2 2 2 22 3 4 5x y x y

2 2 2 24 4 6 9 8 16 10 25x x y y x x y y 2 2 2 24 4 6 9 8 16 10 25x x y y x x y y

12 16 28x y

3 4 7x y