Coordinate Graphs of Triangles

The illustrated problems below are graphs of the following triangles:

• isosceles triangles
• scalene triangles
• right triangles

Concepts Illustrated By the Problems

• distance between two points
• angle between two lines
• point of intersection of two straight lines
• equation of a straight line

Description of the Problems

For each problem, the following information are given:

• equations of the line segments
• slopes of the line segments
• x-intercepts and y-intercepts of the lines
• distances between the terminal points of the line segments
• interior angles of the triangles

Using one or a combination of two or more of the above information, it is possible for a problem to be presented in several ways.

Line segments and their terminal points referred to in the problems are the sides and vertices, respectively, of the triangles.

Slope of a line segment with terminal points A and B is abbreviated as m_AB.

Distance between points A and B is abbreviated as distance_AB

Graphs of Isosceles Triangles

Show that points A(-5, 1), B(3, 5), C(2, -3) are vertices of an isosceles triangle.



Distance Between Two Points

• distance_AB = √ 80
• distance_BC = distance_AC = √ 65

Equations of the Line Segments

• line segment AB: x - 2y + 7 = 0 (x-intercept = -7; y-intercept = 7/2)
• line segment BC: 8x - y - 19 = 0 (x-intercept = 19/8, y-intercept = -19)
• line segment AC: 4x + 7y + 13 = 0 (x-intercept = - 13/4, y-intercept = - 13/7)

Slopes of the Line Segments

• m_AB = 1/2
• m_BC = 8
• m_AC = - 4/7

Interior Angles of the Triangle

• angle A = angle B = 56.3°
• angle C = 67.4°

Show that points A(-5, 4), B(3, 3), C(-1, -4) are vertices of an isosceles triangle.



Distance Between Two Points

• distance_AB = distance_BC = √ 65
• distance_AC = √ 80

Equations of the Line Segments

• line segment AB: x + 8y - 27 = 0 ( x-intercept = 27, y-intercept = 27/8)
• line segment BC: 7x - 4y - 9 = 0 (x-intercept = 9/7, y-intercept = - 9/4)
• line segment AC: 2x + y + 6 = 0 (x-intercept = -3, y-intercept = -6)

Slopes of the Line Segments

• m_AB = - 1/8
• m_BC = 7/4
• m_AC = -2

Interior Angles of the Triangle

• angle A = angle C = 56.3°
• angle B = 67.4°

Prove analytically that points A(-5, -5), B(-2, 6), C(5, -3) are vertices of an isosceles triangle.



Distance Between Two Points

• distance_AB = distance_BC = √ 130
• distance_AC = √ 104

Equations of the Line Segments

• line segment AB: 11x - 3y + 40 = 0 (x-intercept = - 40/11, y-intercept = 40/3)
• line segment BC: 9x + 7y - 24 = 0 (x-intercept = 24/9, y-intercept = 24/7)
• line segment AC: x - 5y - 20 = 0 (x-intercept = 20, y-intercept = -4)

Slopes of the Line Segments

• m_AB = 11/3
• m_BC = - 9/7
• m_AC = 1/5

Interior Angles of the Triangle

• angle A = angle C = 63.4°
• angle B = 53.1°

Graphs of Scalene Triangles

Show that points A(-4, 1), B(1, 0), C(-2, -5) are vertices of a scalene triangle.



Distance Between Two Points

• distance_AB = √ 26
• distance_BC = √ 34
• distance_AC = √ 40

Equations of the Line Segments

• line segment AB: x + 5y - 1 = 0 (x-intercept = 1, y-intercept = 1/5)
• line segment BC: 5x - 3y - 5 = 0 (x-intercept = 1, y-intercept = - 5/3)
• line segment AC: 3x + y + 11 = 0 (x-intercept = - 11/3, y-intercept = -11)

Slopes of the Line Segments

• m_AB = - 1/5
• m_BC = 5/3
• m_AC = -3

Interior Angles of the Triangle

• angle A = 60.3°
• angle B = 70.3°
• angle C = 49.4°

Show that the following points are vertices of a scalene triangle:

• A (-5, -4)
• B (3, 4)
• C (4, -2)

Distance Between Two Points

• distance_AB = √ 128
• distance_BC = √ 37
• distance_AC = √ 85

Equations of the Line Segments

• line segment AB: x - y + 1 = 0 (x-intercept = -1, y-intercept = 1)
• line segment BC: 6x + y - 22 = 0 (x-intercept = 11/3, y-intercept = 22)
• line segment AC: 2x - 9y - 26 = 0 (x-intercept = 13, y-intercept = - 26/9)

Slopes of the Line Segments

• m_AB = 1
• m_BC = -6
• m_AC = 2/9

Interior Angles of the Triangle

• angle A = 32.5°
• angle B = 54.5°
• angle C = 93.0°

Show that the points A(-3, 5), B(5, -2), C(-1, -5) are vertices of a scalene triangle.



Distance Between Two Points

• distance_AB = √ 113
• distance_BC = √ 45
• distance_AC = √ 104

Equations of the Line Segments

• line segment AB: 7x + 8y - 19 = 0 (x-intercept = 19/7, y-intercept = 19/8)
• line segment BC: x - 2y - 9 = 0 (x-intercept = 9, y-intercept = - 9/2)
• line segment AC: 5x + y + 10 = 0 (x-intercept = -2, y-intercept = -10)

Slopes of the Line Segments

• m_AB = - 7/8
• m_BC = 1/2
• m_AC = -5

Interior Angles of the Triangle

• angle A = 37.5°
• angle B = 67.8°
• angle C = 74.7°

Graphs of Right Triangles

Prove that points A(-1, 5), B(5, -1), C(-4, 2) are vertices of a right triangle.

Distance Between Two Points

• distance_AB = √ 72
• distance_BC = √ 90
• distance_AC = √ 18

Equations of the Line Segments

• line segment AB: x + y - 4 = 0 (x-intercept = 4, y-intercept = 4)
• line segment BC: x + 3y - 2 = 0 (x-intercept = 2, y-intercept = 2/3)
• line segment AC: x - y + 6 = 0 (x-intercept = -6, y-intercept = 6)

Slopes of the Line Segments

• m_AB = -1
• m_BC = - 1/3
• m_AC = 1

Interior Angles of the Triangle

• angle A = 90.0°
• angle B = 26.6°
• angle C = 63.4°

Show that points A(-5, 3), B(2, -1), C(-1, -3) are vertices of a right triangle.



Distance Between Two Points

• distance_AB = √ 65
• distance_BC = √ 13
• distance_AC = √ 52

Equations of the Line Segments

• line segment AB: 4x + 7y - 1 = 0 (x-intercept = 1/4, y-intercept = 1/7)
• line segment BC: 2x - 3y - 7 = 0 (x-intercept = 7/2, y-intercept = - 7/3)
• line segment AC: 3x + 2y + 9 = 0 (x-intercept = -3, y-intercept = - 9/2)

Slopes of the Line Segments

• m_AB = - 4/7
• m_BC = 2/3
• m_AC = - 3/2

Interior Angles of the Triangle

• angle A = 26.6°
• angle B = 63.4°
• angle C = 90.0°

Prove that points A(-2, 5), B(4, -3), C(0, -6) are vertices of a right triangle.



Distance Between Two Points

• distance_AB = √ 100
• distance_BC = √ 25
• distance_AC = √ 125

Equations of the Line Segments

• line segment AB: 4x + 3y - 7 = 0 (x-intercept = 7/4, y-intercept = 7/3)
• line segment BC: 3x - 4y - 24 = 0 (x-intercept = 8, y-intercept = -6)
• line segment AC: 11x + 2y + 12 = 0 (x-intercept = - 12/11, y-intercept = -6)

Slopes of the Line Segments

• m_AB = - 4/3
• m_BC = 3/4
• m_AC = - 11/2

Interior Angles of the Triangle

• angle A = 26.6°
• angle B = 90.0°
• angle C = 63.4°

Graphs of Isosceles Right Triangles

Prove that the points A(-4, 1), B(1, 5), C(0, -4) are vertices of an isosceles right triangle.



Distance Between Two Points

• distance_AB = √ 41
• distance_BC = √ 82
• distance_AC = √ 41

Equations of the Line Segments

• line segment AB: 4x - 5y + 21 = 0 (x-intercept = - 21/4, y-intercept = 21/5)
• line segment BC: 9x - y - 4 = 0 (x-intercept = 4/9, y-intercept = -4)
• line segment AC: 5x + 4y + 16 = 0 (x-intercept = - 16/5, y-intercept = -4)

Slopes of the Line Segments

• m_AB = 4/5
• m_BC = 9
• m_AC = - 5/4

Interior Angles of the Triangle

• angle A = 90°
• angle B = 45°
• angle C = 45°

Show that points A(-5, 4), B(0, 2), C(-2, -3) are vertices of an isosceles right triangle.



Distance Between Two Points

• distance_AB = √ 29
• distance_BC = √ 29
• distance_AC = √ 58

Equations of the Line Segments

• line segment AB: 2x + 5y - 10 = 0 (x-intercept = 5, y-intercept = 2)
• line segment BC: 5x - 2y + 4 = 0 (x-intercept = - 4/5, y-intercept = 2)
• line segment AC: 7x + 3y + 23 = 0 (x-intercept = - 23/7, y-intercept = - 23/3)

Slopes of the Line Segments

• m_AB = - 2/5
• m_BC = 5/2
• m_AC = - 7/3

Interior Angles of the Triangle

• angle A = 45°
• angle B = 90°
• angle C = 45°

Prove that points A(-3, 4), B(5, 1), C(-6, -4) are vertices of an isosceles right triangle.



Distance Between Two Points

• distance_AB = √ 73
• distance_BC = √ 146
• distance_AC = √ 73

Equations of the Line Segments

• line segment AB: 3x + 8y - 23 = 0 (x-intercept = 23/3, y-intercept = 23/8)
• line segment BC: 5x - 11y - 14 = 0 (x-intercept = 14/5, y-intercept = - 14/11)
• line segment AC: 8x - 3y + 36 = 0 (x-intercept = - 9/2, y-intercept = 12)

Slopes of the Line Segments

• m_AB = - 3/8
• m_BC = 5/11
• m_AC = 8/3

Interior Angles of the Triangle

• angle A = 90°
• angle B = 45°
• angle C = 45°