Question

A dog pen has the shape of a right-angled triangle with sides of length 30, 40, and 50. a straight fence goes from the corner with the right-angle to a point on the opposite side, dividing the dog pen into two sections which have the same perimeter. how long is the fence?

Answer 1

The length of the fence that divides a right-angled triangular dog pen into two sections with equal perimeters is 10 units long.

Step-by-step explanation:

The question asks us to find the length of a fence that divides a right-angled triangular dog pen into two sections with equal perimeters. The sides of the triangle are 30, 40, and 50 units, which makes it a Pythagorean triple, confirming that the triangle is indeed right-angled.

If we denote the vertices of the triangle as A, B, and C where C is the right angle, and the fence cuts the hypotenuse (AB) at point D, then the perimeter of triangle ACD will be equal to AD + CD + CA, and the perimeter of triangle CBD will be BD + CD + CB.

To find the length of AD (which is also the length of the fence), we set the perimeters of triangles ACD and CBD equal to each other. This gives us AD + CD + CA = BD + CD + CB, knowing that CA = 30, CB = 40, and AB = 50. This equation simplifies to AD + 30 = (50 - AD) + 40. Solving for AD, we find that AD = 10 units. Hence, the fence that divides the dog pen into two sections with equal perimeters is 10 units long.