If the diagonals of a trapezoid are congruent, then the trapezoid is isosceles.

Question

Thanks in advance to anyone who can help me out on this. I'm currently a junior in high school taking and doing well my school's honors pre-calc class, but of all of the math I've ever learned, proofs have always given me trouble. And that's what we've just begun re-learning, my kryptonite. Anyways, I just can't figure out how to prove this problem like it says.

Prove that if the diagonals of a trapezoid are congruent, then the trapezoid is isosceles, using coordinate geometry.

I'm solely restricted to using things like the midpoint formula, distance formula, slope formula, etc. I can't use any theorems from geometry other than the Pythagorean Theorem.

I've tried drawing a trapezoid with the points (0,0), (a, b), (a+c, b), and (d,0), and then since the given is that the diagonals are equal, finding the distances of the diagonals and setting them equal, and solving them for one variable. Then I took that variable and then plugged it into the distances of the two legs of the trapezoid, since an isosceles trapezoid has two congruent legs. Unfortunately, I just can't get the two legs of the trapezoid to end up being congruent. If anyone can explain how to solve this problem, that would be awesome!

Answer 1

Hint: Let the vertices, in counterclockwise order, be $P,Q,R,S$, with $PQ$ parallel to $RS$.

Let $P=(0,0)$, $Q=(a,0)$, $R=(b,h)$, and $S=(c,h)$.

Because the diagonals are equal, you know that $b^2+h^2=(c-a)^2+h^2$.

You want to show that the two non-parallel sides are equal, so you want to show that $c^2+h^2=(b-a)^2+h^2$.

Answer 2

Ok, here is one approach that works. Since diagonals are equal, and bases are parallel, it is easy to show that top and bottom triangle are similar, using alternate interior and vertical angles theorem. Let's call the diagonal intersection point S and vertices of the trapezoid A,B,C,D with A at bottom left vertex. Now BS is a multiple of DS (say factor k) and AS is a multiple of CS (same factor k) Now use area formula for triangle ½*b*c*sin(enclosed angle) on left and right triangle to show that these areas are equal. You will in both cases arrive at area ½*AS*DS*sin(angle) for left and right triangle of the trapezoid. KNowing that these triangles are congruent, and top and bottom triangles are similar, I think you can finish the proof