The figure shows an isosceles trapezoid. Given that 𝐹𝐾 is equal to nine and 𝐽𝐺 is equal to 26, find 𝐾𝐻.
So we’ll begin by putting the information in the question onto the diagram. 𝐹𝐾 is equal to nine. And 𝐽𝐺 is equal to 26. Notice as well that 𝐽𝐺 is a diagonal of the trapezoid. It connects a pair of opposite corners. The key word in the question is that this trapezoid is isosceles. Which means that the pair of nonparallel sides are congruent to each other. So we know that 𝐹𝐽 and 𝐺𝐻 are the same length. However, we’ve been asked to find 𝐾𝐻 which is part of the diagonal 𝐹𝐻 of the trapezoid.
What do we know about the diagonals of isosceles trapezoids? Well, in fact there’s a really key fact about them which is that they are congruent. This isn’t true for trapezoids in general. But it is true when the trapezoid in question is isosceles, as ours is here. So in this question, this means that 𝐹𝐻 is equal to 𝐽𝐺 as they are the two diagonals of the trapezoid. We can divide the diagonal 𝐹𝐻 up into the two portions on either side of the point 𝐾 where the two diagonals meet. So we have 𝐹𝐾 plus 𝐾𝐻 is equal to 𝐽𝐺.
Now we know the lengths of two of these line segments as they’re given in the question. 𝐹𝐾 is nine and 𝐽𝐺 is 26. We can substitute them into our equation given nine plus 𝐾𝐻 is equal to 26. Remember, it’s 𝐾𝐻 that we’re looking to find. So now we have a straightforward equation that we can use. To find 𝐾𝐻, we just need to subtract nine from both sides.
This tells us that 𝐾𝐻 is equal to 17. So 17 is our answer to the problem. Remember, the key piece of information that we used in this question was that the diagonals of an isosceles trapezoid are congruent.