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Friday, June 02, 2023
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You have a right triangle divided into two smaller triangles. The height of all three triangles is not given, but you know it's the same height for all three. The object is to figure out the area of the blue shaded region. We know that the large triangle's hypotenuse has a length of 34, and the interior line segment has a length of 20. We also know the length of the segment between the big triangle's right angle and the point where the interior line segment intersects with the large triangle's bottom leg measures 12.
ReplyDeleteWe need to calculate the triangle's height. The small left-hand triangle has a hypotenuse of 20 and a leg of 12. Using the Pythagorean theorem, we realize that all three triangles have a height of 16 (because this is a 3:4:5 right triangle with side ratios of 12:16:20, 3•4:4•4:5•4).
But what's the length of the large triangle's bottom leg? We know it's 12 plus something. The big triangle's hypotenuse is 34, so we can again use Pythagoras:
16^2 + (x + 12)^2 = 34^2
256 + (x + 12)^2 = 1156
(x + 12)^2 = 900
This reduces to
x + 12 = 30, or
x = 18.
The height of the blue shaded triangle is 16, and we now know the base is 18.
A = (1/2)bh
A = (1/2)(18)(16)
A = 144
QED.