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Geometry and Trigonometry Difficulty: Hard

The line segment shown in the xy-plane represents one of the legs of a right triangle. The area of this triangle is $36 StartRoot 13 EndRoot$ square units. What is the length, in units, of the other leg of this triangle?

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Explanation

Choice B is correct. The length of a segment in the xy-plane can be found using the distance formula, $\sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$ , where $left parenthesis x 1 comma y 1 right parenthesis$ and $left parenthesis x 2 comma y 2 right parenthesis$ are the endpoints of the segment. The segment shown has endpoints at $left parenthesis negative 6 comma 4 right parenthesis$ and $left parenthesis 3 comma 10 right parenthesis$ . Substituting $left parenthesis negative 6 comma 4 right parenthesis$ and $left parenthesis 3 comma 10 right parenthesis$ for $left parenthesis x 1 comma y 1 right parenthesis$ and $left parenthesis x 2 comma y 2 right parenthesis$ , respectively, in the distance formula yields $\sqrt{{(3 - (- 6))}^{2} + {(10 - 4)}^{2}}$ , or $StartRoot 9 squared plus 6 squared EndRoot$ , which is equivalent to $StartRoot 81 plus 36 EndRoot$ , or $StartRoot 117 EndRoot$ . Let $x$ represent the length, in units, of the other leg of this triangle. The area, $A$ , of a right triangle can be calculated using the formula $A = \frac{1}{2} b h$ , where $b$ and $h$ are the lengths of the legs of the triangle. It's given that the area of the triangle is $36 StartRoot 13 EndRoot$ square units. Substituting $36 StartRoot 13 EndRoot$ for $A$ , $StartRoot 117 EndRoot$ for $b$ , and $x$ for $h$ in the formula $A = \frac{1}{2} b h$ yields $36 StartRoot 13 EndRoot equals one half left parenthesis StartRoot 117 EndRoot right parenthesis left parenthesis x right parenthesis$ . Multiplying both sides of this equation by $2$ yields $72 StartRoot 13 EndRoot equals x StartRoot 117 EndRoot$ . Dividing both sides of this equation by $StartRoot 117 EndRoot$ yields $\frac{72 \sqrt{13}}{\sqrt{117}} = x$ . Multiplying the numerator and denominator on the left-hand side of this equation by $StartRoot 117 EndRoot$ yields $\frac{72 \sqrt{1,521}}{117} = x$ , or $\frac{72 (39)}{117} = x$ , which is equivalent to $\frac{2,808}{117} = x$ , or $24 equals x$ . Therefore, the length, in units, of the other leg of this triangle is $24$ .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. $3 StartRoot 13 EndRoot$ is equivalent to $StartRoot 117 EndRoot$ , which is the length, in units, of the line segment shown in the xy-plane, not the length, in units, of the other leg of the triangle.

Choice D is incorrect and may result from conceptual or calculation errors.