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Right triangles

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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?

Apr 24, 2022

#1
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How many square units are in the area of triangle APC?

Hello Guest!

$$AN(x)=-x+6\\ MC(x)=-\frac{3}{12}x+3\\ -x+6=-\frac{1}{4}x+3\\ \color{blue}x=4$$

$$A_{APC}=A_{ABC}-A_{BCM}-A_{AMP}\\ A_{APC}=\dfrac{12\cdot 6}{2}-\dfrac{12\cdot 3}{2}-\dfrac{3\cdot {\color{blue}4}}{2}\\ \color{blue}A_{APC}=12$$

!

Apr 24, 2022
edited by asinus  Apr 24, 2022
#2
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The area of the total triangle is $$6 \times 12 \div 2 = 36$$

Draw Median $$BO$$. All 6 triangles have an area of 6

$$\triangle APC$$ consists of exactly 2 of these triangles, meaning the area is $$2 \times 6 = \color{brown}\boxed{12}$$

Apr 24, 2022