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

Guest Jul 19, 2023

#1**0 **

Upload the picture of your triangle. If they don't have one, then learn to draw one and uploaded.

Guest Jul 19, 2023

#3**0 **

*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*

We don't need all those midpoints and angle bisectors and whatever MY is.

We have the three sides of a triangle, and that's all we need to find its area.

We could still get the area, but it's an isoceles triangle, and that makes it easier.

The sides: PQ = 36, PR = 36, QR = 16

This tells us that triangle PQR is an isoceles triangle, with angle P at the apex.

So drop a perpendicular line from P down to base QR, intersect at point J.

This does two things. It bisects QR and it creates two right triangles.

So use Pythagoras' formula to determine the length of this perpendicular,

which, conveniently for us, is also the height of PQR.

PJ^{2} = 36^{2} – 8^{2} -----> PJ^{2} = 1296 – 64 -----> PJ = 35**.**1

Area of PQR = (1 / 2) • base • height = (1 / 2) • 16 • 35**.**1 = **280.8**

__check answer__

per Heron's Formula, Area = sqrt of (44)(8)(8)(28) = sqrt(78,848) = **280.8**

Confirmed. I love it when a plan comes together.

_{.}

Bosco Jul 20, 2023