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1. In the diagram below, points A, B, C, and P are situated so that PA=2, PB=3, PC=4, and BC=5. What is the maximum possible area of triangle ABC?

Diagram: https://latex.artofproblemsolving.com/f/8/c/f8c4039f30c9ec08fd75e8d55b476ded54ceb800.png (background is transparent)

2. In triangle ABC, \angle ABC = 90^\circ and AD is an angle bisector. If AB=90, BC=x, and AC=2x-6, then find the area of triangle ABC. Round your answer to the nearest integer.

3. Medians $$\overline{DP}$$ and $$\overline{EQ }$$ of triangle DEF are perpendicular. If $$DP=18$$ and $$EQ=24$$, then what is $$DF$$

Jan 25, 2019
edited by yasbib555  Jan 25, 2019

#1
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3. Medians DP and EQ of triangle DEF are perpendicular....DP = 18  and  EQ =  24

Let the medians intersect at   M

Then   DM = (2/3)DP =  (2/3)18 =  12

And EM =  (1/3)EQ =  (1/3)(24) = 8

So.....using the Pythagorean Theorem, ......DQ  =   sqrt ( DM^2 + EM^2 )  =  sqrt (12^2 + 8^2)  =  sqrt (144 + 64) =

sqrt ( 208)  =  4sqrt (13)

But  DF = 2DQ =   2 * 4sqrt (13)  =   8sqrt (13)   Jan 25, 2019
#2
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thank you so much!!

yasbib555  Jan 26, 2019
#3
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2. In triangle ABC, \angle ABC = 90° and AD is an angle bisector. If AB=90, BC=x, and AC=2x-6, then find the area of triangle ABC. Round your answer to the nearest integer.

Don't see where the angle bisector comes into play, here.....

We have that

BC^2 + AB^2 = AC^2

x^2 + 90^2 =  (2x - 6)^2

x^2 + 90^2 =  4x^2 - 24x + 36

3x^2 - 24x  - 8064 = 0

x^2 - 8x - 2688 = 0    this factors as

(x - 56) (x + 48) = 0

x = 56

So....the area is   (1/2)product of leg lengths  = (1/2)(AB)(BC) = (1/2)(90)(56) = 2520 units^2   Jan 26, 2019