Exercise 23 Page 650

We do not have any information to prove congruence between any of the triangle's angles. Therefore, we need to prove congruence by the SSS Congruence Theorem. To calculate the distance of a side, we can use the Distance Formula. d=sqrt((x_2-x_1)^2+(y_2-y_1)^2) However, this isn't necessary for all sides. From the diagram, we see that the triangles share OQ as a side. By the Reflexive Property of Congruence, we already know this side is congruent. Additionally, one side in each triangle is vertical. The length of a vertical side is the absolute value of the difference of the endpoints y-coordinates.

The remaining two sides, RQ and PO, we have to calculate using the Distance Formula.

d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)

SubstitutePoints

Substitute ( h,k) & ( 0,0)

d_(PO) = sqrt(( h - 0)^2 + ( k - 0)^2)

SubTerms

Subtract terms

d_(PO) = sqrt(h^2 + k^2)

We will do the same thing for RQ.

d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)

SubstitutePoints

Substitute ( h,k+j) & ( 0,j)

d_(RQ) = sqrt(( h - 0)^2 + ( k+j - j)^2)

SubTerms

Subtract terms

d_(RQ) = sqrt(h^2 + k^2)

The length of the triangle's third side is the same as well. Since the three sides of △ OPQ are congruent to the three sides of △ QRO, we can by the SSS Congruence Theorem prove that △ OPQ ≅ △ QRO.