Exercise 63 Page 529

We are asked to prove that the diagonals of an isosceles trapezoid are congruent. We are asked to use a coordinate proof, so let's place the trapezoid in the coordinate plane so that one base is on the x-axis and one vertex is at the origin.

Since side BC is parallel to the x-axis, the y-coordinates of B and C are the same. We are asked to investigate the length of the diagonals, so let's recall the Distance Formula. 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 to express the length of the legs of the trapezoid.

Points	Substitution	Length
A(0,0) and B(b,p)	sqrt((b-0)^2+(p-0)^2)	AB=sqrt(b^2+p^2)
C(c,p) and D(d,0)	sqrt((d-c)^2+(0-p)^2)	CD=sqrt((d-c)^2+p^2)

Since the trapezoid is isosceles, we know that AB=CD. Let's substitute the expression from the table and simplify the resulting equation.

AB=CD

SubstituteExpressions

Substitute expressions

sqrt(b^2+p^2)=sqrt((d-c)^2+p^2)

RaiseEqn

LHS^2=RHS^2

b^2+p^2=(d-c)^2+p^2

SubEqn

LHS-p^2=RHS-p^2

b^2=(d-c)^2

Since d>c, we can remove the exponents. b=d-c We can substitute this for the first coordinate of B.

Let's use the Distance Formula again to express the length of the diagonals.

Points	Substitution	Length
A(0,0) and C(c,p)	sqrt((c-0)^2+(p-0)^2)	AC=sqrt(c^2+p^2)
B(d-c,p) and D(d,0)	sqrt((d-(d-c))^2+(0-p)^2)	BD=sqrt(c^2+p^2)

We can see that AC=BD, so the diagonals of an isosceles trapezoid are indeed congruent.