Question

there are two right triangles with a common vertex (the intersection of their hypotenuses). one right triangle has a leg of length 3.5 and a hypotenuse of length 7. the other right triangle has a leg of length 4 and a hypotenuse of length x. we need to find the value of x.

Explanation:

Step1: Identify similar triangles

The two right triangles are similar (vertical angles and right angles imply AA similarity). So, the ratios of corresponding sides are equal.

Step2: Set up proportion

Let the sides be: for the smaller triangle (right) with leg 3.5 and hypotenuse 7, and the larger triangle (left) with leg 4 and hypotenuse \( x \). The ratio of leg to hypotenuse should be equal: \( \frac{3.5}{7}=\frac{4}{x} \)

Step3: Solve for \( x \)

Cross - multiply: \( 3.5x = 7\times4 \)
\( 3.5x=28 \)
Divide both sides by 3.5: \( x=\frac{28}{3.5}=8 \)

Answer:

\( x = 8 \)