Coordinate Proofs

Explain why it is convenient to place a right triangle on the grid as shown when writing a coordinate proof.

• The hypotenuse of the right triangle is easy to identify.

• The side lengths are often easier to find because you are using zeros in your expressions.

• It is easier to dilate the figure on the coordinate plane.

• Both legs have the same length when you place the triangle on the x- and y-axes.

Write a plan for the proof.

Given: G is the midpoint of HF
Prove: △GHJ≅△GFO

• Find the coordinates of G using the Midpoint Formula
  Use these coordinates and the Distance formula to show that OG ≅ JG.
  Show that HG≅ FG by the definition of midpoint and ∠HGJ ≅ ∠FGO by the Vertical Angles Congruence Theorem.
  

• Find the coordinates of G using the Distance Formula
  Use these coordinates and the Midpoint formula to show that OG ≅ JG.
  Show that HG≅ FG by the definition of midpoint and ∠HGJ ≅ ∠FGO by the Vertical Angles Congruence Theorem.

• Then, use the SAS Congruence Theorem to conclude that △GHJ ≅ △GFO.

• Then, use the SSS Congruence Theorem to conclude that △GHJ ≅ △GFO.

You and your cousin are camping in the woods. You hike to a point that is 500 meters east and 1200 meters north of the campsite. Your cousin hikes to a point that is 1000 meters east of the campsite.

The distance from you to your cousin is ______ meters.

• 1500

• 1300

• 1200

• 1000

How is a coordinate proof different from other types of proofs you have studied?

• You do not need to write a plan for a coordinate proof.

• You do not have a Given or Prove statement.

• You have to assign coordinates to vertices and write expressions for the side lengths and slopes of segments.

• You can only do coordinate proofs with triangles.

Lamar is writing a coordinate proof to show that a segment from the midpoint of the hypotenuse of a right triangle to the opposite vertex forms two triangles with equal areas. He starts by assigning coordinates as given.
N is the midpoint of segment KL. Therefore, the coordinates of N are (a, ___)

• ab

• b

• 2a

• 4b

Complete the coordinate proof of the theorem.
Given: ABCD is a square.
Prove: The diagonals of ​ ABCD ​ are perpendicular.
The coordinates of square ABCD A(0, 0), B(____, 0), C(_____, a), and ​ D(0, a).

• 5 , 4

• a, a

• 1 , -1

• a, 1

Complete the coordinate proof of the theorem.
Given: ABCD is a square.
Prove: The diagonals of ​ ABCD ​ are perpendicular.
The slope of segment BD when simplified is equal to -1.
The product of the slopes is equal to

• 360

• 0

• 1

• -1

Using coordinate geometry proofs, prove that a quadrilateral with vertices at (0,0), (4,0), (4,3), and (0,3) is a rectangle.

• By showing all angles are 90 degrees.

• By showing opposite sides are parallel and equal in length.

• By showing diagonals bisect each other.

• By showing it has four sides.

T (___, ___)

• (0, 2a)

• (2a, 2a)

• (2a, y)

• (2a, 0)

C (___, q)