Right Triangles in Coordinate Plane

1. 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.

2. 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.

Select the graph that represents an isosceles right triangle with leg length p in the most convenient way.

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Select the most convenient graph to represent a scalene triangle with one side length of 2m.

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Write a plan for the proof.

Given Coordinates of vertices of △OPM and △ONM

Prove △OPM and △ONM are isosceles triangles

• Find the lengths of OP, PM, MN, NO and OM to show that △OMP≅△OMN by the SSS Congruence Theorem.

• Find the lengths of OP, PM, MN, and NO to show that OP ≅ PM and MN ≅ NO.

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.

Find the coordinates of the vertex O.

Find the coordinates of the vertex U.

• (k, 0)

• (0, k)

• (k, 2k)

• (2k, k)

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 the campsite to you is ______ meters.

• 1500

• 1300

• 1200

• 1000

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