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
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. AX2 = PX2 + RX2 AX2 = 24.922 + 182 AX2 = 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.