Question

find the length x.

Explanation:

Step1: Identify similar triangles

The two right triangles are similar (AA similarity, right angle and vertical angles). So, corresponding sides are proportional.

Step2: Set up proportion

Let the sides be: for the first triangle, hypotenuse 8, leg \( x \); for the second, hypotenuse 7, leg 3.5. Wait, no—correct correspondence: the segments of the hypotenuse and the legs. Wait, the hypotenuse is split into 8 and 7? No, wait, the two triangles: one has leg \( x \), hypotenuse segment 8? Wait, no, the diagram: two right triangles sharing a common angle (vertical angles), so similar. So the ratio of corresponding legs is equal to the ratio of corresponding hypotenuse segments? Wait, no, let's label: let the big triangle (left) have legs \( x \) and (let's say) the other leg, and hypotenuse part 8. The small triangle (right) has legs 3.5 and the other leg, hypotenuse part 7. Wait, actually, the correct proportion for similar right triangles (geometric mean theorem) states that in a right triangle, the altitude to the hypotenuse is the geometric mean, but here it's two similar triangles. Wait, the two triangles: one with hypotenuse segment 8 and leg \( x \), the other with hypotenuse segment 7 and leg 3.5. Wait, no, the correct proportion is \( \frac{x}{3.5} = \frac{8}{7} \)? Wait, no, let's see: the two triangles are similar, so corresponding sides. Let’s denote the first triangle (left) has legs \( x \) and (let's say) \( a \), hypotenuse \( 8 + 7 = 15 \)? No, the diagram shows two right triangles, one with leg \( x \), hypotenuse segment 8, and the other with leg 3.5, hypotenuse segment 7. Wait, actually, the correct proportion is \( \frac{x}{3.5} = \frac{8}{7} \)? Wait, solving: cross - multiply \( 7x = 3.5\times8 \)

Step3: Solve for x

\( 7x = 28 \), so \( x=\frac{28}{7}=4 \)

Answer:

\( 4 \)