Triangle T is a right triangle with a 12-inch hypotenuse and an area of 28 square inches. What is the perimeter, in inches, of triangle T? (A) 20 (B) 28 (C) 12 + sqrt(210) (D) 32 (E) 45

Here it goes: you have the area of 28. That means that (c1*c2)/2=28, so 2c1*c2=4*28=112. Hyp is 12, so (c1)^2+(c2)^2 = 12^2 = 144. Since (c1+c2)^2= (c1)^2+(c2)^2 +2c1*c2, (c1+c2)^2 = 256, which is 16^2. So c1+c2 = 16. Perimeter = c1+c2+hyp= 16+12=28

Awesome work! Here's my solution as well: Let x and y be the two legs of the triangle. A 12-inch hypotenuse means that x^2 + y^2 = 12^2 (144) An area of 28 means that 1/2xy=28 or xy=56 or (most importantly) 2xy=112 Note: We aren’t required to find the value x and the value of y. We need only the sum x+y to determine the entire perimeter. Important part: Notice that when we expand (x+y)^2 = x^2 + 2xy + y^2, we get some recognizable equalities from above. Let’s perform some substitutions: (x+y)^2 = x^2 + 2xy + y^2 =(x^2 + y^2) + 2xy = 144 + 112 = 256 If (x+y)^2 = 256, then x+y = 16 and so the perimeter is 16+12=28 (Answer = B)

What about the 45-45-90 triangle rule. Divide 12 by sq. rt. 2 = roughly 8 2 (8) + 12 = roughly 28 (I actually got 8.5 for the legs and 29 for perimeter, but figured this was due to approximations and picked 28 since it was the closest to 29) Is that an acceptable way to solve the problem or was it just a coincidence that it worked?

Great question Brent. Would you mind telling us the source?

canada_sms wrote:Great question Brent. Would you mind telling us the source? Hi fellow Canadian, I have been creating my own questions for my posts.

Since this is a right triangle, the sides would be in the ratio of 3:4:5. The hypotenuse being the largest side solve for 7:5 with 5 as 12. This will give you 84/5. (84/5)+20 = 28 4/5 and the closest answer would be B.

arzanr wrote:Since this is a right triangle, the sides would be in the ratio of 3:4:5. T WOW!! A right angle triangle can have INFINITE number of variations and 3,4,5 is JUST one of them. Watch out! 5,12 and 13 for example.

You are correct, not all right triangles would be 3:4:5, but a triangle with the ratio 3:4:5 would be a right triangle. Thanks for making that important clarification!