Video: Finding the Diagonal Length of a Trapezoid given Its Other Diagonal Length and Other Dimensions

The figure shows an isosceles trapezoid. Given that ๐น๐พ = 9 and ๐ฝ๐บ = 26, find ๐พ๐ป.

02:21

Video Transcript

The figure shows an isosceles trapezoid. Given that ๐น๐พ is equal to nine and ๐ฝ๐บ is equal to 26, find ๐พ๐ป.

So weโ€™ll begin by putting the information in the question onto the diagram. ๐น๐พ is equal to nine. And ๐ฝ๐บ is equal to 26. Notice as well that ๐ฝ๐บ 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 ๐น๐ฝ and ๐บ๐ป are the same length. However, weโ€™ve been asked to find ๐พ๐ป which is part of the diagonal ๐น๐ป of the trapezoid.

What do we know about the diagonals of isosceles trapezoids? Well, in fact thereโ€™s a really key fact about them which is that they are congruent. This isnโ€™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 ๐น๐ป is equal to ๐ฝ๐บ as they are the two diagonals of the trapezoid. We can divide the diagonal ๐น๐ป up into the two portions on either side of the point ๐พ where the two diagonals meet. So we have ๐น๐พ plus ๐พ๐ป is equal to ๐ฝ๐บ.

Now we know the lengths of two of these line segments as theyโ€™re given in the question. ๐น๐พ is nine and ๐ฝ๐บ is 26. We can substitute them into our equation given nine plus ๐พ๐ป is equal to 26. Remember, itโ€™s ๐พ๐ป that weโ€™re looking to find. So now we have a straightforward equation that we can use. To find ๐พ๐ป, we just need to subtract nine from both sides.

This tells us that ๐พ๐ป 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.

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