A rectangular park has two walking paths as shown. If 𝑃𝑆 equals 100 metres and 𝑃𝑅 equals 213 metres, find the length of line segment 𝑄𝑇.
Let’s have a look at this rectangle and fill in the lengths that we’re given. We’re told that 𝑃𝑆 is 100 metres and 𝑃𝑅 is 213 metres. it might be very tempting at this point to look at our values and think we may have to use the Pythagorean theorem, since we did after all have 90-degree angles in this rectangle. However, this question is asking us to find the length of 𝑄𝑇, which is marked in blue on the diagram. Notice that the line 𝑃𝑅 and the line 𝑄𝑆 are the diagonals of the rectangle. So let’s see if we can use any facts about the diagonals of the rectangle to help.
We can recall that the diagonals of a rectangle are congruent, which means that they’re exactly the same length. They also bisect each other, which means that each diagonal cuts the other one into two identical length pieces. So if we look at our rectangle, then the diagonal 𝑄𝑆 is equal to the other diagonal 𝑃𝑅, since they are congruent. And therefore they’re both 213 metres. And so to find the length of 𝑄𝑇, we know that since our diagonals bisect each other, then this is equal to half of the length 𝑄𝑆. So we find half of 213 which is 106.5 metres. So our answer for the length of 𝑄𝑇 is 106.5 metres. In this question, we were given the information that 𝑃𝑆 is 100 metres which we didn’t need to use. This is common in exam questions to distract us from the information that we do need to use.