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 between the intersecting lines). So, their corresponding sides are proportional.

Step2: Set up proportion

Let the sides of the first triangle (with legs 8 and 6) correspond to the sides of the second triangle (with legs 6 and \(x\)). The proportion is \(\frac{8}{6}=\frac{6}{x}\)? Wait, no, wait. Wait, actually, the first triangle has legs 8 and 6 (the horizontal leg is 6, vertical is 8), the second has vertical leg 6 and horizontal leg \(x\). Wait, no, maybe I mixed up. Wait, the two triangles: one has legs 8 (vertical) and 6 (horizontal), the other has legs 6 (vertical) and \(x\) (horizontal). Wait, no, actually, the triangles are similar, so the ratio of vertical to horizontal should be equal. Wait, first triangle: vertical leg 8, horizontal leg 6. Second triangle: vertical leg 6, horizontal leg \(x\). Wait, no, that can't be. Wait, maybe the triangles are similar with the first triangle's vertical leg 8, horizontal leg 6, and the second triangle's vertical leg 6, horizontal leg \(x\). Wait, no, actually, the correct proportion is \(\frac{8}{x}=\frac{6}{6}\)? No, that's not right. Wait, let's look again. The two triangles: one has legs 8 (vertical) and 6 (horizontal), the other has legs 6 (vertical) and \(x\) (horizontal). Wait, no, the angles: the right angle, and the angle between the hypotenuse and the horizontal line is common. So, the triangles are similar, so the ratio of vertical leg to horizontal leg should be equal. So, for the first triangle (left), vertical leg is 8, horizontal leg is 6. For the second triangle (right), vertical leg is 6, horizontal leg is \(x\). Wait, no, that would be \(\frac{8}{6}=\frac{6}{x}\)? No, that gives \(8x = 36\), \(x = 4.5\), but that seems off. Wait, maybe I mixed up the legs. Wait, the left triangle: vertical leg 8, horizontal leg 6. The right triangle: vertical leg 6, horizontal leg \(x\). Wait, no, maybe the horizontal leg of the left is 6, vertical is 8; the right triangle has vertical leg 6, horizontal leg \(x\). Wait, but maybe the proportion is \(\frac{8}{6}=\frac{6}{x}\)? Wait, no, that's inverse. Wait, actually, the correct proportion is \(\frac{8}{6}=\frac{6}{x}\)? No, let's use similar triangles properly. Let’s denote the left triangle as \(\triangle ABC\) with \( \angle B = 90^\circ \), \( AB = 8 \), \( BC = 6 \). The right triangle as \(\triangle DEF\) with \( \angle E = 90^\circ \), \( DE = 6 \), \( EF = x \). Since \(\triangle ABC \sim \triangle DEF\) (AA similarity: right angle and common angle), then \(\frac{AB}{DE}=\frac{BC}{EF}\). So, \( \frac{8}{6}=\frac{6}{x} \)? Wait, no, \( AB \) is vertical, \( DE \) is vertical; \( BC \) is horizontal, \( EF \) is horizontal. So, \( \frac{AB}{DE}=\frac{BC}{EF} \) => \( \frac{8}{6}=\frac{6}{x} \). Wait, solving for \(x\): \( 8x = 6 \times 6 \) => \( 8x = 36 \) => \( x = \frac{36}{8} = 4.5 \)? Wait, that seems low. Wait, maybe I got the legs reversed. Maybe the left triangle has horizontal leg 6, vertical leg 8, and the right triangle has horizontal leg \(x\), vertical leg 6. So, the ratio of vertical to horizontal for left is \( \frac{8}{6} \), for right is \( \frac{6}{x} \). Wait, no, that would be \( \frac{8}{6} = \frac{6}{x} \), which is what I did. But let's check again. Wait, maybe the triangles are similar with the first triangle's horizontal leg 6, vertical leg 8, and the second triangle's horizontal leg \(x\), vertical leg 6. So, the proportion is \( \frac{8}{6} = \frac{6}{x} \), so \( x = \frac{6 \times 6}{8} = \frac{36}{8} = 4.5 \), which is \( \frac{9}{2} \) or 4.5. Wait, but let's confirm. Alternatively, maybe the triangles are similar with the left triangle's legs 8 and 6, and the right triangle's legs \(x\) and 6. Wait, no, the right triangle's vertical leg is 6, horizontal is \(x\). The left triangle's vertical is 8, horizontal is 6. So, similarity ratio: 8/6 = 6/x. So, x = (6*6)/8 = 36/8 = 9/2 = 4.5. So, x is 4.5, which is 9/2 or 4.5.

Answer:

\( \boxed{4.5} \) (or \( \frac{9}{2} \))