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 = 16, and MY = 50, then find the area of triangle PQR
+42 To find the area of triangle PQR, we need to calculate the lengths of PX and YQ.
Using the Pythagorean theorem, we can calculate the length of PX:
PX^2 = PR^2 - PQ^2 = 1296 - 1296 = 0
PX = 0
Using the Law of Cosines, we can calculate the length of YQ:
YQ^2 = QR^2 + MY^2 - 2*QR*MY*cos(QPR)
YQ^2 = 256 + 2500 - 2*16*50*cos(QPR)
YQ^2 = 2808
YQ = 53.08
Now, we can calculate the area of triangle PQR using Heron's formula:
A = sqrt(s*(s-PQ)*(s-PR)*(s-QR))
where s is the semiperimeter of the triangle
s = (PQ + PR + QR)/2
s = (36 + 36 + 16)/2 = 88/2 = 44
A = sqrt(44*(44-36)*(44-36)*(44-16))
A = sqrt(44*8*8*28)
A = sqrt(2808*28)
A = 280.8
Correct yes, GPT also yes. Why do you bother with all of the unecessary information when you can just use heron's directly?
"To find the area of triangle PQR, we need to calculate the lengths of PX and YQ.
Using the Pythagorean theorem, we can calculate the length of PX:
PX^2 = PR^2 - PQ^2 = 1296 - 1296 = 0
PX = 0
Using the Law of Cosines, we can calculate the length of YQ:
YQ^2 = QR^2 + MY^2 - 2*QR*MY*cos(QPR)
YQ^2 = 256 + 2500 - 2*16*50*cos(QPR)
YQ^2 = 2808
YQ = 53.08"
Also I'm pretty sure PX isn't zero. This is also how GPT writes math. You never use YQ or PX later, another sign of GPT.
