Triangles and Coordinate Proofs

Chapter 5: Congruent Triangles

Section 5.8: Coordinate Proofs

Coordinate Proofs

Definition

A coordinate proof
involves placing a
geometric figures in a
coordinate plane.

Numbers vs. Variables

When you use numbers,
the work is only valid for
that specific figure.

When you use variables
for the coordinates, the
results are true for all
figures of that type.

Example 1: Placing a Figure in a Coordinate Plane

Place each figure in a coordinate plane in a way that is convenient for finding side
lengths. Assign coordinates to each vertex.

a)

A rectangle

b) a triangle

Example 1 Solution

Example 2: Applying Variable Coordinates

Place an isosceles right triangle in a coordinate plane. Then find the length of
the hypotenuse and the coordinates of its midpoint M.

Example 2 Solution

Example 3: Writing a Plan for a Coordinate Proof

Example 3 Solution

Multiple Choice

A square has vertices (0,0), (m,0), and (0,m). What is the fourth vertex?

• (0, 2m)

• (m, m)

• (n, 0)

• (m, n)

Multiple Choice

A triangle has vertices O (0, 0), H (m, n), and J (m, 0). What kind of triangle is ΔOHJ?

• Acute

• Obtuse

• Right

Multiple Choice

What expression represents the length of $$\overline{OJ}$$ ?

• m

• n

• $$\sqrt[]{m^2+n^2}$$

Multiple Choice

What expression represents the length of $$\overline{HJ}$$ ?

• m

• n

• $$\sqrt[]{m^2+n^2}$$

Multiple Choice

What expression represents the length of $$\overline{OH}$$ ?

• m

• n

• $$\sqrt[]{m^2+n^2}$$

Multiple Choice

What expression represents the midpoint of $$\overline{OJ}$$ ?

• $$\left(m,\frac{n}{2}\right)$$

• $$\left(\frac{m}{2},0\right)$$

• $$\left(\frac{m}{2},\frac{n}{2}\right)$$

Multiple Choice

What expression represents the midpoint of $$\overline{HJ}$$ ?

• $$\left(m,\frac{n}{2}\right)$$

• $$\left(\frac{m}{2},0\right)$$

• $$\left(\frac{m}{2},\frac{n}{2}\right)$$

Multiple Choice

What expression represents the midpoint of $$\overline{OH}$$ ?

• $$\left(m,\frac{n}{2}\right)$$

• $$\left(\frac{m}{2},0\right)$$

• $$\left(\frac{m}{2},\frac{n}{2}\right)$$

Multiple Choice

What formula could be used to show that $$\overline{GO}\cong\overline{GH}$$ ?

• Midpoint

• Distance

• Slope

Multiple Choice

If $$\overline{GJ}$$ bisects $$\angle OGH$$ , which two angles are congruent?

• $$\angle GOJ\cong\angle GHJ$$

• $$\angle GJO\cong\angle GJH$$

• $$\angle JGO\cong\angle JGH$$

Fill in the Blanks

$\overline{GJ}\cong\overline{GJ}$ by the

Fill in the Blanks

What shortcut can be used, in conjunction with the previous questions, to show that $\Delta GJO\cong\Delta GJH$ ?

Example 4: Writing a Coordinate Proof

Solution to Example 4

Example 5: Modeling in Real Life

Solution to Example 5