Question

a right triangle has one leg that is 2 meters longer than the other leg. if the hypotenuse is 10 meters, find the lengths of the legs.

Explanation:

Step1: Define variables

Let the length of the shorter leg be \( x \) meters. Then the length of the longer leg is \( x + 2 \) meters.

Step2: Apply Pythagorean theorem

For a right triangle, \( a^2 + b^2 = c^2 \), where \( a \) and \( b \) are the legs and \( c \) is the hypotenuse. So we have \( x^2+(x + 2)^2=10^2 \).

Step3: Expand and simplify the equation

Expand \( (x + 2)^2 \): \( x^2+x^2 + 4x+4 = 100 \). Combine like terms: \( 2x^2+4x + 4-100 = 0 \), which simplifies to \( 2x^2+4x-96 = 0 \). Divide through by 2: \( x^2+2x - 48 = 0 \).

Step4: Solve the quadratic equation

Factor the quadratic: \( (x + 8)(x - 6)=0 \). Set each factor equal to zero: \( x+8 = 0 \) or \( x - 6 = 0 \). Solving these, we get \( x=-8 \) (discard since length can't be negative) or \( x = 6 \).

Step5: Find the lengths of the legs

The shorter leg \( x = 6 \) meters. The longer leg is \( x + 2=6 + 2 = 8 \) meters.

Answer:

The lengths of the legs are 6 meters and 8 meters.