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In the picture below, which is not drawn to scale, △ABC is right-
angled at C. The two legs AC and BC have length 40 and 60. The
shaded region consists of all points inside △ABC which are at a
distance less than or equal to 6 from one (or both) of the two legs of
△ABC. What is the area of the shaded region?
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These ones are real stumpers for me. Sorry, but I can't help much here.
|dw:1328767721354:dw|We can divide up the shaded area into the three sections I outlined in the picture. The two triangle area we know are similar to the larger one, so we can find their side lengths and thus their area.
Now that we know their side-lengths, we can find the length/width of the rest of the shaded area and since it's just two rectangles stuck together, we can easily find the area.
My final answer would be 525. You can check that if you want.
yep, that's correct!
But how would you find the side length?
These are the two triangle you have to construct at the top, and bottom respectively:|dw:1328768286221:dw|Using some ratios, we know that the one on the top has a bottom length of 6, and since it's similar to the larger one, the other side length must be 4. Likewise, for the small triangle on the bottom, the vertical height must be 6, so again since the triangle is similar to the larger one, the bottom length must be 9. From this, we can find the area of the two triangles easily.
Now, we need to find the area of the rest of the shaded area. If we just remove the small triangles, we get something with a constant width of 6, and it's (40 -4) high, and (60-9) wide. So we don't overlap, I just let the bottom rectangle be (51-6) wide instead of 51. The you add (36*6 + 45*6) to the area of the triangles, and this gets 525.