Question

question
solve for ( x ). leave your answer in simplest radical form.
(there is a diagram of two right triangles sharing a common leg. the left triangle has legs 6 and the common leg, hypotenuse 9. the right triangle has legs 4 and ( x ), and the common leg.)
answer attempt 1 out of 3
( x = ) input box ( sqrt{,,} ) submit answer

Explanation:

Step1: Find the height of the left triangle

Let the height of the left right - triangle (with hypotenuse 9 and base 6) be \( h \). Using the Pythagorean theorem \( a^{2}+b^{2}=c^{2} \), where \( c = 9 \) and \( a = 6 \), we have \( h^{2}+6^{2}=9^{2} \).
So \( h^{2}=9^{2}-6^{2}=81 - 36=45 \).

Step2: Use the height to find \( x \) in the right triangle

In the right - triangle on the right (with hypotenuse equal to \( h \) and legs 4 and \( x \)), by the Pythagorean theorem \( x^{2}+4^{2}=h^{2} \).
We know that \( h^{2} = 45 \), so \( x^{2}+16 = 45 \).
Then \( x^{2}=45 - 16=29 \).
Taking the square root of both sides, since \( x>0 \) (length), we get \( x=\sqrt{29} \).

Answer:

\( \sqrt{29} \)