In triangle PQR, M is the midpoint of PQ. Let X be the point on QR such that PX bisects angle QPR, and let the perpendicular bisector of PQ intersect AX at Y. If PQ = 36, PR = 36, QR = 36, and MY = 36, then find the area of triangle PQR

Guest Jul 17, 2023

#1**0 **

We first try to find the length of PM, so we divide PQ by 2. PM = PQ/2 = 36/2 = 18

Then, we use the bisector theorem to find the lengths of PX and QX. Because PX bisects the angle QPR, PX / RX = PQ / RQ

Through subsitution, we can get 36/26, or 18/13.

PX = (18 * 18) / 13 = 24.92, and RX = (13 * 18) / 13 = 18

Using the Pythagorean theorem, we can find AX. AX^{2} = PX^{2} + RX^{2} AX^{2} = 24.92^{2} + 18^{2 }AX^{2} = 621 +324 AX = sqrt945 AX = 30.74.

Because Y is on the perpendicular bisector of PQ, PY = QY = PQ / 2 = 18

Therefore, AY = AX - XY = 30.74 - 8, or 22.74

Finally, we can use Heron's formula to find the area of the triangle PQR. s = (36 + 22 + 26) / 2 = 42

area(PQR) = sqrt(s(s-36)(s-22)(s-26)) = sqrt(42*6*20*16) = 336

Therefore, the area of triangle PQR is **336** square units.

history Jul 17, 2023