Question

find the length x.

Explanation:

Step1: Identify similar triangles

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

Step2: Set up proportion

Let the sides be: for the smaller triangle, legs 3 and 6; for the larger, leg \( x \) and hypotenuse 7? Wait, no, wait. Wait, the sides: the smaller triangle has leg 3, hypotenuse 6? Wait, no, the right triangle: one has leg 3, hypotenuse 6? Wait, no, the two triangles: one has leg 3, and the other has leg \( x \); the hypotenuses? Wait, no, the sides: the triangles are similar, so \(\frac{3}{x}=\frac{6}{7}\)? Wait, no, wait. Wait, the triangles: the first triangle (right) has leg 3, and the other side (the one opposite the vertical angle) is 6? Wait, no, the diagram: two right triangles, with a common vertical angle. So, the angles are equal (right angle and vertical angle), so triangles are similar. So, the sides: for the first triangle (let's say triangle A), legs: 3 and the other leg (let's say 6? Wait, no, the diagram: one triangle has leg 3, and the hypotenuse? Wait, no, the labels: one triangle has leg 3, and the side adjacent to the vertical angle is 6; the other triangle has leg \( x \), and the side adjacent to the vertical angle is 7. Wait, no, maybe the sides are: the two triangles have corresponding sides. So, the ratio of corresponding legs is equal to the ratio of corresponding hypotenuses? Wait, no, in similar triangles, corresponding sides are proportional. So, if one triangle has legs \( a, b \) and hypotenuse \( c \), the other has legs \( ka, kb \) and hypotenuse \( kc \). Wait, but in the diagram, one triangle has leg 3, and the other leg (the one forming the vertical angle) is 6? Wait, no, the labels: 3, 6, 7, \( x \). So, let's assume that the triangles are similar, so \(\frac{3}{x}=\frac{6}{7}\)? Wait, no, cross - multiply: \( 6x = 3\times7\), \( 6x = 21\), \( x=\frac{21}{6}=\frac{7}{2}=3.5 \). Wait, no, maybe the proportion is \(\frac{3}{6}=\frac{x}{7}\). Yes, that makes sense. Because the corresponding legs: 3 and \( x \), and the other corresponding sides: 6 and 7? Wait, no, the sides: the triangle with leg 3 has the other side (the one opposite the vertical angle) as 6, and the triangle with leg \( x \) has the other side as 7. So, the ratio of the legs: \(\frac{3}{x}=\frac{6}{7}\)? Wait, no, AA similarity: right angle and vertical angle. So, the sides: the first triangle (let's call it triangle 1) has legs \( 3 \) and \( 6 \) (wait, no, 3 is a leg, 6 is the hypotenuse? No, 3 and 6 are legs? Wait, no, 3 is a leg, and 6 is the other leg? Wait, the diagram: two right triangles, intersecting at the vertical angle. So, one triangle has legs 3 and 6, the other has legs \( x \) and 7? No, that can't be. Wait, maybe the sides are: the triangle with leg 3 has hypotenuse 6, and the triangle with leg \( x \) has hypotenuse 7? No, hypotenuse must be longer than leg. 6 is longer than 3, 7 is longer than \( x \). So, similar triangles: \(\frac{3}{x}=\frac{6}{7}\). Solving for \( x \): \( 6x = 3\times7\), \( x=\frac{21}{6}=\frac{7}{2}=3.5 \). Wait, but let's check again. The two triangles are similar, so corresponding sides are proportional. So, the leg of the first triangle (3) corresponds to the leg of the second (\( x \)), and the other leg of the first (6) corresponds to the other leg of the second (7). So, \(\frac{3}{x}=\frac{6}{7}\). Cross - multiply: \( 6x = 21\), so \( x=\frac{21}{6}=\frac{7}{2}=3.5 \) or \( \frac{7}{2} \).

Answer:

\(\frac{7}{2}\) (or 3.5)