Quadrilateral 𝑃𝑄𝑅𝑆 has vertices 𝑃 at negative one, six; 𝑄 at five, six; 𝑅 at five, three; and 𝑆 at negative one, three. Find the length of 𝑄𝑆.
𝑄𝑆 cuts our quadrilateral into two triangles, and we can actually use either of those triangles to find the length of 𝑄𝑆. Let’s go ahead and choose triangle 𝑄𝑅𝑆. And at angle 𝑅, that’s a 90-degree angle. How do we know that?
Well, segment 𝑄𝑅 is vertical because our 𝑥-coordinate doesn’t change. So it just goes directly up from three to six and stays at five for 𝑥. 𝑆𝑅 is horizontal because from 𝑆 to 𝑅, we don’t rise at all. We stay at three and we go from negative one to five for our 𝑥. So it runs left and right. That means these are perpendicular. Therefore, angle 𝑅 is a right angle. And since we have a right angle at 𝑅, that means triangle 𝑄𝑅𝑆 is a right triangle and we can use the Pythagorean theorem.
The Pythagorean theorem is useful because it will allow us to find 𝑄𝑆. Because the Pythagorean theorem states that 𝑎 squared plus 𝑏 squared equals 𝑐 squared, where 𝑎 and 𝑏 are the two shorter sides and 𝑐 is the longest side — the side across from the 90-degree angle. And because we have this diagram, we actually know the side lengths of 𝑎 and 𝑏 because we can count them.
From 𝑅 to 𝑄, we went up three spaces. So 𝑎 is three. You can also look at our points: 𝑅 is at five, three and 𝑄 is at five, six. So 𝑥 didn’t change at all, but our 𝑦 is changed. And it changed from three to six; that’s a difference of three. Now looking at 𝑆𝑅, from 𝑆 to 𝑅, we can count the spaces and find that it’s six. Or looking at our points, we stayed at three for 𝑦, but we changed for 𝑥 — from negative one to five. And the difference between these numbers is six. So let’s go ahead and fill in this information into our Pythagorean theorem.
So we have three squared plus six squared equals 𝑐 squared, and 𝑐 will be the side 𝑄𝑆 that we wanna find. So three squared is nine, six squared is 36. Bring down your equals 𝑐 squared. Now, we need to add nine plus 36. So 45 equals 𝑐 squared. In order to solve for 𝑐, we need to get rid of the squared. The inverse operation of squaring a number would be to square root a number. So let’s square root both sides.
So 𝑐 is equal to the square root of 45; however, this can be simplified. 45 is five times nine. The reason we’re doing this is because nine is a perfect square. So you can actually take that out of the square root and the square root of nine is three. Three times three equals nine. So three will come outside of the square root and five will be left inside of the square root, also known as the radical sign. So 𝑐 is equal to three square root five. Therefore, the length of 𝑄𝑆 is three square root five. 𝑄𝑆 equals three square root five.