Questions: Find the unknown length in the right triangle. Simplify the answer if possible. The unknown length c in the triangle is (Simplify your answer. Type an exact answer, using radicals as needed.)

Solution Steps

To find the unknown length in a right triangle, 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 sides. If one side is given as 12 and the other side is unknown, we can solve for the unknown side using this theorem.

In the right triangle, we have one side length \( a = 12 \) and the hypotenuse \( c = 90 \).

According to the Pythagorean theorem, we can express the relationship between the sides of the triangle as: \[ c^2 = a^2 + b^2 \] where \( b \) is the unknown side length we want to find.

To find \( b \), we rearrange the equation: \[ b^2 = c^2 - a^2 \]

Substituting the known values into the equation: \[ b^2 = 90^2 - 12^2 \]

Calculating the squares: \[ b^2 = 8100 - 144 = 7956 \] Now, taking the square root to find \( b \): \[ b = \sqrt{7956} \approx 89.1964 \]

Final Answer