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# right triangle ABC

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In right triangle ABC, M and N are midpoints of legs AB and BC, respectively. Leg AB is 6 units long, and leg BC is 12 units long. How many square units are in the area of triangle APC?

Dec 16, 2020

#1
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In the right triangle, ABC, M, and N are midpoints of legs AB and BC, respectively. Leg AB is 6 units long, and leg BC is 12 units long. How many square units are in the area of triangle APC?

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[AMP] ≅ [CPM]

[NBMP] ≅ [APC]

[AMP] : [CPM] : [NBMP] : [APC] = 1 : 1 : 2 : 2

[APC] = 1/3 [ABC] = 36 / 3

Dec 17, 2020
#3
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Instead of [CPM] it should be [CNP]    ( two places )

Guest Dec 17, 2020
#2
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Let

A = (0,6)   M  = (0,3)   N = (6,0)    C = (12,0)

Slope of AN =  ( 6 - 0) / (0 - 6)  =  -1

Equation of  line  containing  AN .....   y =   -1x + 6

Slope of MC  = ( 0-3)/(12 - 0)  =  -3/12   = -1/4

Equation of  line  containing MC .....     y = (-1/4)x+ 3

x coordinate  of P  = altitude of triangle AMP

(-1/4)x + 3 =  -1x + 6

(3/4) x  = 3

x = 4

And [ AMP ]  =  (1/2) AM * 4  = 2 * 3   =  6

And  area of triangle AMC  = (1/2) AM BC  = (1/2) (3) (12)    =18

So area of  APC =  [ AMC ] - [ AMP] =   18  - 6    = 12

Dec 17, 2020