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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 = 22, QR = 26, and MY = 8, then find the area of triangle PQR

Jul 4, 2023

#1
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Jul 4, 2023
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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 = 22, QR = 26, and MY = 8, then find the area of triangle PQR

I was trying to draw the figure, and then

you threw me a curve when you said AX.

Where did that come from?

Then I saw that you have the values of all

three sides.  You don't need anything else.

Add all three sides, then take half of it.

Call this "s".

36 + 22 + 26

s  =  ——————  =  42

2

Then use the following to get the area.         A  =  sqrt[(s)(s – a)(s – b)(s – c)]

This is called Heron's Formula.

A  =  sqrt[(42)(42 – 36)(42 – 22)(42 – 26)]

A  =  sqrt(42 • 6 • 20 • 16)

A  =  sqrt(80,640)

A  =  283.97 square units

That answer looks weird,but it has

to be right ... it's Heron's Formula.

.

Jul 4, 2023
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First we  denote the length of PX as 'x'. By applying the Angle Bisector Theorem, we have:

PR / RQ = PX / XQ

22 / 26 = x / (26 - x)

22 * (26 - x) = 26x

572 - 22x = 26x

48x = 572

x ≈ 11.92

Now, let's find the length of XQ:

XQ = QR - PX ≈ 26 - 11.92 ≈ 14.08

Since M is the midpoint of PQ, we can determine the length of PM:

PM = PQ / 2 = 36 / 2 = 18

Since MY is perpendicular to PQ and MY = 8, we can determine the length of PY:

PY = PM - MY = 18 - 8 = 10

Therefore, AY is equal to PY:

AY = PY = 10

Now, we have all the side lengths of triangle AXY. We can calculate its area using Heron's formula:

s = (XY + AY + AX) / 2 = (14.08 + 10 + 11.92) / 2 ≈ 18

Area_AXY = sqrt(s * (s - XY) * (s - AY) * (s - AX))

Area_AXY = sqrt(18 * (18 - 14.08) * (18 - 10) * (18 - 11.92))

Area_AXY ≈ sqrt(18 * 3.92 * 8 * 6.08) ≈ 24.48

Finally, the area of triangle PQR is twice the area of triangle AXY:

Area_PQR = 2 * Area_AXY ≈ 2 * 24.48 ≈ 48.96 square units

Jul 5, 2023
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sorry, but none of these are correct...

thanks for trying though....

I still need help

Jul 5, 2023
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Hii @icecreamlover,

Sorry, I cant' help you for it. Now I interested for this questions correct answer. If you find it then share with me. myfedloan org bill

Jul 6, 2023
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I was able to find the answer...it's   88

icecreamlover  Jul 8, 2023
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heron's formula--->√42x6x20x16--->√80640--->48√35

Jul 6, 2023