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Big Ideas Math Integrated I, 2016

BI

Big Ideas Math Integrated I, 2016 View details

8. Coordinate Proofs

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Exercise 18 Page 644

Use the Midpoint Formula to calculate the coordinates of the midpoints.

See solution.

Practice makes perfect

Consider the given information and the desired outcome of the proof. Given:& Coordinates of△ DEA, & H is the midpoint ofDA, & Gis the midpoint of EA. Prove:& DG≅ EH& Let's begin by taking a look at the given diagram.

Since H and G are the midpoints of DA and AE, respectively, we can use the Midpoint Formula to find their coordinates.

M(x_1+x_2/2,y_1+y_2/2)
Midpoint	Endpoints	Substitute	Simplify
H	D( - 2h,0) and A( 0,2k)	H(- 2h+ 0/2,0+ 2k/2)	H(- h,k)
G	E( 2h,0) and A( 0,2k)	G(2h+ 0/2,0+ 2k/2)	G(h,k)

Let's show this in our diagram.

Now that we know the coordinates of the midpoints, we can calculate the length of DG and EH using the Distance Formula.

d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2)
Segment	Endpoints	Substitute	Simplify
EH	E( 2h,0) and H( - h,k)	EH=sqrt(( - h - 2h)^2+( k- 0)^2)	EH=sqrt(9h^2+k^2)
DG	G( - 2h,0) and D( h,k)	DG=sqrt(( h-( - 2h))^2+( k- 0)^2)	DG=sqrt(9h^2+k^2)

Both segments have the same length. Therefore, by the definition of congruent segments, we can conclude that DG≅ EH.

Subchapter links

Communicate Your Answer p.639

Monitoring Progress p.640-642

Exercises p.643-644