The line segment slants gradually up from left to right. The line segment begins at the point (negative 6 comma

Question

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?

Accepted Answer

Choice B is correct. The length of a segment in the xy -plane can be found using the distance formula, x 2 - x 1 2 + y 2 - y 1 2 , where x 1 , y 1 and x 2 , y 2 are the endpoints of the segment. The segment shown has endpoints at - 6 , 4 and 3 , 10 . Substituting - 6 , 4 and 3 , 10 for x 1 , y 1 and x 2 , y 2 , respectively, in the distance formula yields 3 - - 6 2 + 10 - 4 2 , or 9 2 + 6 2 , which is equivalent to 81 + 36 , or 117 . 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 = 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 13 square units. Substituting 36 13 for A , 117 for b , and x for h in the formula A = 1 2 b h yields 36 13 = 1 2 117 x . Multiplying both sides of this equation by 2 yields 72 13 = x 117 . Dividing both sides of this equation by 117 yields 72 13 117 = x . Multiplying the numerator and denominator on the left-hand side of this equation by 117 yields 72 1,521 117 = x , or 72 39 117 = x , which is equivalent to 2,808 117 = x , or 24 = 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 13 is equivalent to 117 , 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.

Question c682f4