Difference between revisions of "1953 AHSME Problems/Problem 33"
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| + | ==Problem== | ||
The perimeter of an isosceles right triangle is <math>2p</math>. Its area is: | The perimeter of an isosceles right triangle is <math>2p</math>. Its area is: | ||
<math>\textbf{(A)}\ (2+\sqrt{2})p \qquad \textbf{(B)}\ (2-\sqrt{2})p \qquad \textbf{(C)}\ (3-2\sqrt{2})p^2\\ \textbf{(D)}\ (1-2\sqrt{2})p^2\qquad \textbf{(E)}\ (3+2\sqrt{2})p^2</math> | <math>\textbf{(A)}\ (2+\sqrt{2})p \qquad \textbf{(B)}\ (2-\sqrt{2})p \qquad \textbf{(C)}\ (3-2\sqrt{2})p^2\\ \textbf{(D)}\ (1-2\sqrt{2})p^2\qquad \textbf{(E)}\ (3+2\sqrt{2})p^2</math> | ||
| + | ==Solution== | ||
Given leg length <math>x</math>, we can write the perimeter of this triangle to be <math>2x+x\sqrt{2}=2p</math>. Thus, <math>x(2+\sqrt{2})=2p</math>. Divide to get <math>x=\frac{2p}{2+\sqrt{2}}</math>. Multiply by the conjugate and simplify to get <math>x=p(2-\sqrt{2})</math>. Square and divide by two to get the area of the triangle, or <math>\frac{p^2*(2-\sqrt{2}^2)}{2}</math>, or <math>p^2(3-2\sqrt{2})</math>. <math>\boxed{C}</math> | Given leg length <math>x</math>, we can write the perimeter of this triangle to be <math>2x+x\sqrt{2}=2p</math>. Thus, <math>x(2+\sqrt{2})=2p</math>. Divide to get <math>x=\frac{2p}{2+\sqrt{2}}</math>. Multiply by the conjugate and simplify to get <math>x=p(2-\sqrt{2})</math>. Square and divide by two to get the area of the triangle, or <math>\frac{p^2*(2-\sqrt{2}^2)}{2}</math>, or <math>p^2(3-2\sqrt{2})</math>. <math>\boxed{C}</math> | ||
| + | |||
| + | ==See Also== | ||
| + | {{AHSME 50p box|year=1953|num-b=32|num-a=34}} | ||
| + | |||
| + | {{MAA Notice}} | ||
Latest revision as of 00:40, 4 February 2020
Problem
The perimeter of an isosceles right triangle is 
. Its area is:
Solution
Given leg length 
, we can write the perimeter of this triangle to be 
. Thus, 
. Divide to get 
. Multiply by the conjugate and simplify to get 
. Square and divide by two to get the area of the triangle, or 
, or 
. 
See Also
| 1953 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 32 |
Followed by Problem 34 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
