Question

A right-angle triangle has perimeter equivalent to 7 times its height, which is the shortest side, while the area of the triangle is 17.5 times the length of the shortest side. Find the measures of the three sides of the triangle.

• A. 28, 45, 53
• B. 20, 21, 29
• C. 12, 35, 37
• D. 7, 24, 25

Hint:

In problems involving right-angle triangles, use the perimeter, area, and Pythagorean theorem together to find the sides. Always substitute and simplify step by step.

Answer 1

The Correct Option is C

Let the sides of the right triangle be \( a \), \( b \) (where \( a \) is the shortest side), and \( c \) (the hypotenuse). Step 1: Perimeter condition.
We are given that the perimeter of the triangle is 7 times the height. The height of the triangle is the shortest side \( a \), so: \[ \text{Perimeter} = a + b + c = 7a \] This gives the equation: \[ b + c = 6a \quad \text{(1)} \] Step 2: Area condition.
The area of the triangle is given as 17.5 times the shortest side. The area of a right triangle is \( \frac{1}{2}ab \), so: \[ \frac{1}{2}ab = 17.5a \] Simplifying: \[ b = 35 \quad \text{(2)} \] Step 3: Using the Pythagorean theorem.
Since the triangle is a right-angle triangle, we can apply the Pythagorean theorem: \[ a^2 + b^2 = c^2 \] Substitute \( b = 35 \) from equation (2): \[ a^2 + 35^2 = c^2 \quad \text{(3)} \] Step 4: Solve the system of equations.
Substitute \( b = 35 \) into equation (1): \[ 35 + c = 6a \] Solving for \( c \): \[ c = 6a - 35 \quad \text{(4)} \] Substitute equation (4) into equation (3): \[ a^2 + 35^2 = (6a - 35)^2 \] Expanding both sides: \[ a^2 + 1225 = 36a^2 - 420a + 1225 \] Simplify: \[ a^2 = 36a^2 - 420a \] \[ 35a^2 - 420a = 0 \] \[ a(35a - 420) = 0 \] Thus, \( a = 12 \). Step 5: Find the other sides.
Substitute \( a = 12 \) into equation (2): \[ b = 35 \] Substitute \( a = 12 \) into equation (4): \[ c = 6(12) - 35 = 72 - 35 = 37 \] Step 6: Conclusion.
The sides of the triangle are \( a = 12 \), \( b = 35 \), and \( c = 37 \). Therefore, the correct answer is option (3): 12, 35, 37.