Question

In rectangle ANHG, whose perimeter is 100, OP, PQ, and QR are congruent and mutually perpendicular and O is the midpoint of AN. If GH = 40 which is PQ?

Answer 1

Since this is a rectangle, if GH is 40, then AN must be 40.
We know the perimeter is the sum of all the sides, and we have two of the sides (40 and 40). We have that the perimeter is equal to 40+40+AG+NH, and since this is a rectangle, NH and AG must be equal. Lets label them as “x” for now.

So 100 = 40 + 40 + x + x
100 = 80 + 2x
2x = 100 - 80
2x = 20
x = 10

So AG and NH are 10. If we look at the three lines in the rectangle, we may notice that if we were to “stack” OP and QR together, we would get the length of AG and NH (which is 10). Since all three lines are congruent (equal), we can label them all as y.

We know OP and QR are equal, and if their sum makes the length of NH or AG, we can say

AG = 2y
10 = 2y
y = 5

And, again, since theyre all congruent, PQ is equal to both OP and QR, so PQ must be 5.

Answer 2

5

Explanation:

The sum of adjacent sides of the rectangle is half the perimeter, 50, so ...

... AH = 50-40 = 10

Then ...

... OP +QR = 10 = 2×OP . . . . . QR ≅ OP

... OP = 5 = PQ . . . . . . . . . . . . PQ ≅ OP