Question

find the length x.

Explanation:

Step1: Identify Similar Triangles

The two right triangles are similar (by AA similarity, as both have a right angle and share the angle formed by the intersecting lines). So, their corresponding sides are proportional.

Step2: Set Up Proportion

For similar triangles, the ratio of corresponding sides is equal. So, $\frac{8}{6}=\frac{6}{x}$ (wait, no, correct correspondence: the vertical side of the first triangle is 8, horizontal is 6; vertical side of the second is 6, horizontal is x. Wait, actually, the correct proportion is $\frac{8}{6}=\frac{6}{x}$? No, wait, let's check the sides. The first triangle has legs 8 and 6, the second has legs 6 and x. Wait, no, the hypotenuse is common? Wait, no, the triangles are similar, so the ratio of the vertical leg to horizontal leg of first triangle equals ratio of vertical leg to horizontal leg of second triangle. Wait, first triangle: vertical leg 8, horizontal leg 6. Second triangle: vertical leg 6, horizontal leg x. So proportion is $\frac{8}{6}=\frac{6}{x}$? Wait, no, that would be if the angles are corresponding. Wait, actually, the right angles are equal, and the angle between the hypotenuse and the horizontal leg is common? Wait, no, the triangles are similar, so the ratio of the longer leg to shorter leg should be equal. Wait, maybe I mixed up. Let's do it correctly. The two triangles are similar, so the ratio of the vertical side to the horizontal side of the first triangle is equal to the ratio of the vertical side to the horizontal side of the second triangle. Wait, first triangle: vertical = 8, horizontal = 6. Second triangle: vertical = 6, horizontal = x. So $\frac{8}{6}=\frac{6}{x}$? No, that's not right. Wait, maybe the first triangle has legs 8 and 6, the second has legs 6 and x. Wait, no, the right angles are at the bottom, so the first triangle (left) has vertical leg 8, horizontal leg 6. The second triangle (right) has vertical leg 6, horizontal leg x. The hypotenuses are the same line, so the triangles are similar. Therefore, the ratio of vertical leg to horizontal leg of first triangle is equal to ratio of vertical leg to horizontal leg of second triangle. Wait, no, the vertical leg of first is 8, horizontal is 6. Vertical leg of second is 6, horizontal is x. So $\frac{8}{6}=\frac{6}{x}$? Solving for x: $8x = 36$ → $x = \frac{36}{8} = 4.5$? Wait, no, that seems off. Wait, maybe the proportion is $\frac{8}{x}=\frac{6}{6}$? No, that would be x=8, which is not right. Wait, maybe I got the correspondence wrong. Let's label the triangles. Let’s call the left triangle ABC, with right angle at B, AB=8, BC=6. The right triangle DEF, right angle at E, DE=6, EF=x. The hypotenuses AC and DF are the same line, so angle at C (or F) is common. Therefore, triangle ABC ~ triangle DEF by AA (right angle and common angle). Therefore, corresponding sides: AB/DE = BC/EF. So AB=8, DE=6, BC=6, EF=x. So 8/6 = 6/x. Then cross-multiplying: 8x = 36 → x = 36/8 = 4.5. So x=4.5, which is 9/2.

Answer:

$\boxed{4.5}$ (or $\boxed{\dfrac{9}{2}}$)