Pearson Geometry Common Core, 2011
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Pearson Geometry Common Core, 2011 View details
9. Proofs Using Coordinate Geometry
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Exercise 1 Page 416

Practice makes perfect
a Since the angles of a rectangle are right angles, we can place PQRS in the coordinate plane with two sides on the coordinate axes.
b Let's think about the relationship between the coordinates of P, Q, R, and S.
  • Vertex Q is on the y-axis, so the x-coordinate is 0.
  • Vertex S is on the x-axis, so the y-coordinate is 0.
  • Side RS is parallel to the y-axis, so the x-coordinate of R and S are the same.
  • Side QR is parallel to the x-axis, so the y-coordinate of Q and R are the same.

Let's use variables to name the coordinates of Q, R, and S.

These are just example variables, so your variable choices may differ.

c It is given that PQRS is a rectangle, and the claim is that the diagonals are congruent.

2 &Given:&& PQRS is a rectangle &Prove:&& PR≅QS

d To show that two segments are congruent, it is enough to show that they have the same lengths. In a coordinate proof we can use the Distance Formula to express the length of a segment in terms of the coordinates of the endpoints.
The distance between points(x_1,y_1)and(x_2,y_2)is sqrt((x_2-x_1)^2+(y_2-y_1)^2).Let's use this formula first to find the length of diagonal PR.
sqrt((x_2-x_1)^2+(y_2-y_1)^2)
PR=sqrt(( s- 0)^2+( q- 0)^2)
PR=sqrt(s^2+q^2)
Similarly, we can find the length of diagonal QS.
sqrt((x_2-x_1)^2+(y_2-y_1)^2)
QS=sqrt(( s- 0)^2+( 0- q)^2)
QS=sqrt(s^2+(- q)^2)
QS=sqrt(s^2+q^2)
We can see that PR=QS, so the diagonals of rectangle PQRS are indeed congruent.