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# Triangle

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In triangle $PQR,$ $M$ is the midpoint of $\overline{PQ}.$  Let $X$ be the point on $\overline{QR}$ such that $\overline{PX}$ bisects $\angle QPR,$ and let the perpendicular bisector of $\overline{PQ}$ intersect $\overline{PX}$ at $Y.$  If $PQ = 36,$ $PR = 22,$ and $QR = 20,$ then find the area of triangle $PQR.$

Jan 9, 2024

#1
+290
+1

We can find the area of triangle PQR without using any of the above information other than the side lengths of PQ, PR, and QR. Heron's Formula can help us find the area of any triangle given its 3 sides. Heron's Formula  is: A = √ s ( s - PQ ) ( s - PR ) ( s - QR ), where s is the semiperimeter, or the perimeter of the triangle divided by 2.

the semiperimeter of triangle PQR is ( PQ + PR + QR ) / 2. Plugging in the values, we get:

s = ( 36 + 22 + 20 ) / 2

s = 78/2

s = 39

Now that we know the semiperimeter, we can plug in the values, and solve for the area:

A = √ 39 ( 39 - 36 ) ( 39 - 22 ) ( 39 - 20 )

A = √ 39 ( 3 ) ( 17 ) ( 19 )

A = √ 37791

A ≈ 194.4

Jan 9, 2024
edited by DS2011  Jan 9, 2024
edited by DS2011  Jan 9, 2024

#1
+290
+1

We can find the area of triangle PQR without using any of the above information other than the side lengths of PQ, PR, and QR. Heron's Formula can help us find the area of any triangle given its 3 sides. Heron's Formula  is: A = √ s ( s - PQ ) ( s - PR ) ( s - QR ), where s is the semiperimeter, or the perimeter of the triangle divided by 2.

the semiperimeter of triangle PQR is ( PQ + PR + QR ) / 2. Plugging in the values, we get:

s = ( 36 + 22 + 20 ) / 2

s = 78/2

s = 39

Now that we know the semiperimeter, we can plug in the values, and solve for the area:

A = √ 39 ( 39 - 36 ) ( 39 - 22 ) ( 39 - 20 )

A = √ 39 ( 3 ) ( 17 ) ( 19 )

A = √ 37791

A ≈ 194.4

DS2011 Jan 9, 2024
edited by DS2011  Jan 9, 2024
edited by DS2011  Jan 9, 2024
#2
+129829
0

Thx, DS2011   !!!!!

CPhill  Jan 9, 2024