What is the length of each side?

Question

The length of the hypotenuse of a right triangle is 1cm more than triple that of the shorter leg. The length of the longer leg is 1 cm less than triple that of the shorter leg. Find the lengths of the three sides of the triangle

Answer 1

Let the short leg = x

The longer leg = 3x-1

The hypotenuse = 3x+1

 

Use the Pythagorean Theorem:

 

x² + (3x-1)² = (3x+1)²

x² + 9x² - 6x + 1 = 9x² + 6x + 1

x² - 12x = 0

 

Factors to:

 

x(x-12) = 0

x = 0 and 12

 

Since we can't have a triangle with a side = 0 length, x = 12:

• Short Leg = 12
• Longer Leg = 35
• Hypotenuse = 37

 

 

Answer 2

If a is the shorter side, b is the longer, and c is the hypontenuse, then c=3a+1 and b=3a-1. The pythagorean theorem states a²+b²=c², so here that means that a²+(3a-1)²=(3a+1)². if we simplify this out we eventually get a(a-12)=0, meaning a is 0(impossible) or 12. Then we plug back into our original equations, with b=35 and c=37. If we plug back in to check, we see that the equation 122+352=37² holds true.