Questions: The length of one leg of a right triangle is 4 ft. The length of the hypotenuse is 2 ft longer than the other leg. Find the length of the hypotenuse and the other leg. The length of the hypotenuse is The length of the other leg is (Simplify your answers.)

Solution Steps

To solve this problem, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two legs (a and b). We are given that one leg (a) is 4 ft, and the hypotenuse (c) is 2 ft longer than the other leg (b). We can set up the equation \(c = b + 2\) and use the Pythagorean theorem \(a^2 + b^2 = c^2\) to solve for b and c.

Let's define the variables for the problem:

• Let \( a \) be the length of the other leg of the triangle.
• The length of the hypotenuse is \( a + 2 \) ft.
• The length of one leg is given as 4 ft.

For a right triangle, the Pythagorean theorem states: \[ c^2 = a^2 + b^2 \] where \( c \) is the hypotenuse, and \( a \) and \( b \) are the legs of the triangle.

Substituting the known values: \[ (a + 2)^2 = a^2 + 4^2 \]

Expand the left side of the equation: \[ (a + 2)^2 = a^2 + 4a + 4 \]

Substitute back into the equation: \[ a^2 + 4a + 4 = a^2 + 16 \]

Subtract \( a^2 \) from both sides: \[ 4a + 4 = 16 \]

Subtract 4 from both sides: \[ 4a = 12 \]

Divide by 4: \[ a = 3 \]

Substitute \( a = 3 \) back into the expression for the hypotenuse: \[ \text{Hypotenuse} = a + 2 = 3 + 2 = 5 \]

Final Answer