### Video Transcript

The figure shows an isosceles trapezoid. Given that ๐น๐พ is equal to nine and ๐ฝ๐บ is equal to 26, find ๐พ๐ป.

So weโll begin by putting the information in the question onto the diagram. ๐น๐พ is equal to nine. And ๐ฝ๐บ is equal to 26. Notice as well that ๐ฝ๐บ 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 ๐น๐ฝ and ๐บ๐ป are the same length. However, weโve been asked to find ๐พ๐ป which is part of the diagonal ๐น๐ป of the trapezoid.

What do we know about the diagonals of isosceles trapezoids? Well, in fact thereโs a really key fact about them which is that they are congruent. This isnโt 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 ๐น๐ป is equal to ๐ฝ๐บ as they are the two diagonals of the trapezoid. We can divide the diagonal ๐น๐ป up into the two portions on either side of the point ๐พ where the two diagonals meet. So we have ๐น๐พ plus ๐พ๐ป is equal to ๐ฝ๐บ.

Now we know the lengths of two of these line segments as theyโre given in the question. ๐น๐พ is nine and ๐ฝ๐บ is 26. We can substitute them into our equation given nine plus ๐พ๐ป is equal to 26. Remember, itโs ๐พ๐ป that weโre looking to find. So now we have a straightforward equation that we can use. To find ๐พ๐ป, we just need to subtract nine from both sides.

This tells us that ๐พ๐ป 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.