Lesson 4-7 Triangles and Coordinate Proof

• Coordinate proof- uses figures in a coordinate plane and Algebra to prove geometric concepts.

• Placing Figures in the Coordinate Plane1. Use the origin as the vertex or the center of the figure2. Place at least one side of a polygon on an axis3. Keep the figure within the 1st quadrant, if possible4. Use coordinates that make math easy

Use the origin as vertex X of the triangle.

Place the base of the triangle along the positive x-axis.

Position and label right triangle XYZ with leg d units long on the coordinate plane.

X (0, 0) Z (d, 0)

Position the triangle in the first quadrant.

Since Z is on the x-axis, its y-coordinate is 0. Its x-coordinate is d because the base is d units long.

Since triangle XYZ is a right triangle the x-coordinate of Y is 0. We cannot determine the y-coordinate so call it b.

Answer:

X (0, 0) Z (d, 0)

Y (0, b)

Answer:

Position and label equilateral triangle ABC with side w units long on the coordinate plane.

Name the missing coordinates of isosceles right triangle QRS.

Answer: Q(0, 0); S(c, c)

Q is on the origin, so its coordinates are (0, 0). The x-coordinate of S is the same as the x-coordinate for R, (c, ?).

The y-coordinate for S is the distance from R to S. Since QRS is an isosceles right triangle,

The distance from Q to R is c units. The distance from R to S must be the same. So, the coordinates of S are (c, c).

Answer: C(0, 0); A(0, d)

Name the missing coordinates of isosceles right ABC.

Write a coordinate proof to prove that the segment that joins the vertex angle of an isosceles triangle to the midpoint of its base is perpendicular to the base.

Prove:

The first step is to position and label a right triangle on the coordinate plane. Place the base of the isosceles triangle along the x-axis. Draw a line segment from the vertex of the triangle to its base. Label the origin and label the coordinates, using multiples of 2 since the Midpoint Formula takes half the sum of the coordinates.

Given: XYZ is isosceles.

Proof: By the Midpoint Formula, the coordinates of W,

the midpoint of , is

The slope of or undefined. The

slope of is therefore, .

Write a coordinate proof to prove that the segment drawn from the right angle to the midpoint of the hypotenuse of an isosceles right triangle is perpendicular to the hypotenuse.

Proof: The coordinates of the midpoint D are

The slope of is

or 1. The slope of or –1,

therefore .

DRAFTING Write a coordinate proof to prove that the outside of this drafter’s tool is shaped like a right triangle. The length of one side is 10 inches and the length of another side is 5.75 inches.

Proof: The slope of

or undefined. The slope of

or 0, therefore

DEF is a right triangle.

The drafter’s tool is shaped like a

right triangle.

FLAGS Write a coordinate proof to prove this flag is shaped like an isosceles triangle. The length is 16 inches and the height is 10 inches.

C

Proof: Vertex A is at the origin and B is at (0, 10). The x-coordinate of C is 16. The y-coordinate is halfway between 0 and 10 or 5. So, the coordinates of C are (16, 5).

Determine the lengths of CA and CB.

Since each leg is the same length, ABC is isosceles. The flag is shaped like an isosceles triangle.