# Length of hypotenuse of a right triangle of known area in the xy-plane

**Problem**

For triangle *BOA*, *B* is on the *y*-axis, *O* is the origin, and *A* is on the *x*-axis. Point *C*(5, 2) is on the line *AB*. Find the length of *AB* if the area of the triangle is 36 unit^{2}.

A. 24.31 units | C. 13.42 units |

B. 18.30 units | D. 10.80 units |

**Answer Key**

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[ C ]

**Solution**

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$\text{Area} = \frac{1}{2}ab$
$\dfrac{5}{a} + \dfrac{2}{b} = 1$

$36 = \frac{1}{2}ab$

$b = \dfrac{72}{a}$

$\dfrac{x}{a} + \dfrac{y}{b} = 1$

At *C*(5, 2)

$5b + 2a = ab$

$5 \left( \dfrac{72}{a} \right) + 2a = a\left( \dfrac{72}{a} \right)$

$\dfrac{360}{a} + 2a = 72$

$360 + 2a^2 = 72a$

$a^2 - 36a + 180 = 0$

$a = 30 ~ \text{ and } 6$

$b = 2.4 ~ \text{ and } ~ 12 \text{ respectively}$

For *a* = 30 and *b* = 2.4

$L_{AB} = \sqrt{30^2 + 2.4^2} = 30.10 ~ \text{units}$

For *a* = 6 and *b* = 12

$L_{AB} = \sqrt{6^2 + 12^2} = 13.42 ~ \text{units}$

Answer = [ C ]