Coordinate Triangle Proofs

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.

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.

Tags

CCSS.HSG.GPE.B.7

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.

Find the coordinates of the vertex O.

Tags

CCSS.5.G.A.1

Statements Reasons

1. ​ (a)   1. Given

2. ​ (b)   2. Given

3. ​ (c)   3. Given

4. ​ (d)   4. Definition of midpoint

5. ​ (e)   5. AAS

• ∠DBM ≅ ∠FCM

• ∠BDM ≅ ∠CFM

• M is the midpoint of DF

• DM ≅ MF

• △BDM ≅ △CFM

• ∠BMD ≅ ∠CMF

• BM ≅ CM

• BD ≅ CF

• Definition of midpoint

• Definition of bisect

Tags

CCSS.HSG.SRT.B.5

Statements Reasons

1. ​ (a)   1. Given

2. ​ (b)   2. Given

3. ​ (c)   3. Definition of bisect

4. ​ (d)   4. Vertical Angles Theorem

5. ​ (e)   5. ASA

• ∠1 ≅ ∠2

• JG bisects EH at F

• EF ≅ FH

• ∠3 ≅ ∠4

• △EFJ ≅ △HFG

• ∠EJF ≅ ∠HGF

• EJ ≅ GH

• JF ≅ FG

• Definition of bisect

• Definition of midpoint

Tags

CCSS.HSG.SRT.B.5

Find the coordinates of the vertex U.

• (k, 0)

• (0, k)

• (k, 2k)

• (2k, k)

Tags

CCSS.HSG.GPE.B.6

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

Tags

CCSS.8.G.B.8

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

Tags

CCSS.HSG.GPE.B.7

T (a, ___)

• b

• a

• 0

• b/2

Tags

CCSS.HSG.GPE.B.6