Video: Finding the Length of a Side in a Triangle Using the Right Triangle Altitude Theorem

In the following figure, find the length of 𝑃𝑆.

03:39

Video Transcript

In the following figure, find the length of 𝑃𝑆.

So we’re finding the length of this side. Now, there are couple ways to solve this β€” one of them being triangle congruency. Since we have a 90-degree angle right here, this means the other side is 90 degrees because a straight line 𝑆𝑄 is equal to 180 degrees and 90 plus 90 is 180.

So let’s separate this big triangle into two smaller triangles: the polka dot and the stripped. These triangles are actually congruent. But how do we know these are congruent triangles? Well, by side-angle-side. This means that we have two triangles, where we know that two sides and the included angle are equal. If two sides and the included angle of one triangle are equal to the corresponding sides, an angle of another triangle, then the triangles are congruent.

So we know that this side is 16 and this side is 16. So those sides are congruent. They share this side. So this side must be congruent for one triangle as it is to another. And then, the angles between the sides β€” the included angle β€” are both 90 degrees. So they’re congruent. So by side-angle-side, these triangles are congruent. So the corresponding side meaning the one that goes with 𝑃𝑆 is the side 𝑃𝑄 and it’s equal to 25. So if 𝑃𝑄 is equal to 25, 𝑃𝑆 is equal to 25. Therefore, the length of 𝑃𝑆 is 25.

Now, another way to solve this would be using the Pythagorean theorem because we have right triangles. The Pythagorean theorem states the square of the longest side is equal to the sum of the squares of the shorter sides. So let’s assume once again we don’t know the side of 𝑃𝑆. So we need to find the length of 𝑃𝑅 and then use that to find the length of 𝑃𝑆 with the Pythagorean theorem.

So we can find the length of 𝑃𝑅 using the straight triangle. The two shorter sides would be the one that we don’t know: 𝑃𝑅 and 16. And the longer side is always across from the 90-degree angle. It’s called the hypotenuse. So it’s equal to 25. So for the stripped triangle, we have 25 squared equals 16 squared plus π‘₯ squared and π‘₯ represents the length of 𝑃𝑅. 25 squared is equal to 625. 16 squared is equal to 256. And then, we have π‘₯ squared.

so let’s go ahead and subtract 256 from both sides of the equation. So we have that π‘₯ squared is equal to 369. And actually, we don’t need to solve for π‘₯. We can keep it as π‘₯ squared because now with a polka dot triangle, 𝑦 will represent 𝑃𝑆. So that will be the longest side. And then, the shorter sides will be the 16 and the π‘₯, just like we had before.

But we want π‘₯ squared in our equation because it’s the exact same π‘₯ squared as it is for the stripped triangle. So we have that 𝑦 squared is equal to 256 plus 369 because 16 squared was 256 and we plugged in 369 for π‘₯ squared. Adding this together, we get 625. And now square rooting both sides of the equation, we get that 𝑦 is equal to 25.

And 𝑦 represented 𝑃𝑆. So once again, the length of 𝑃𝑆 is equal to 25.

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