Question

find vw.
image of a triangle with points u, x, v; a segment from u to x (length 45), with y on ux (uy=15), and a segment from y to a point w on xv (xw=50). we need to find vw.
vw = blank box

Explanation:

Step1: Identify Similar Triangles

The lines \( UY \) and \( XY \) are parallel (implied by the similar triangle setup), so triangles \( UXV \) and \( YXW \) are similar. The ratio of \( UY \) to \( UX \) is \( \frac{15}{45}=\frac{1}{3} \)? Wait, no, \( UX = 45 \), \( UY = 15 \), so \( XY=45 - 15 = 30 \)? Wait, actually, the ratio of the heights: \( UY = 15 \), \( UX = 45 \), so the ratio of similarity is \( \frac{UY}{UX}=\frac{15}{45}=\frac{1}{3} \)? Wait, no, maybe the big triangle is \( UXV \) with height 45, and the smaller triangle is \( YXW \) with height \( 45 - 15 = 30 \)? Wait, no, the segment \( UY \) is 15, and \( UX \) is 45, so the ratio of the sides of the similar triangles (since \( YW \parallel UV \)) is \( \frac{XY}{UX}=\frac{30}{45}=\frac{2}{3} \)? Wait, maybe better to use the basic proportionality theorem (Thales' theorem). The line \( YW \) is parallel to \( UV \), so \( \frac{XW}{XV}=\frac{XY}{XU} \). Wait, \( XU = 45 \), \( XY = 45 - 15 = 30 \), \( XW = 50 \), let \( VW = x \), so \( XV = 50 + x \). Then \( \frac{50}{50 + x}=\frac{30}{45} \). Simplify \( \frac{30}{45}=\frac{2}{3} \), so \( \frac{50}{50 + x}=\frac{2}{3} \). Cross - multiply: \( 3\times50 = 2\times(50 + x) \), \( 150 = 100 + 2x \), \( 2x = 50 \), \( x = 25 \)? Wait, no, wait, maybe the ratio is \( \frac{UY}{UX}=\frac{15}{45}=\frac{1}{3} \), so the ratio of the bases should be \( \frac{VW}{XV}=\frac{1}{3} \)? No, maybe I got the triangles reversed. Let's re - express: The two triangles are similar, so the ratio of corresponding sides is equal. The height of the big triangle (from \( U \) to \( XV \)) is 45, and the height of the small triangle (from \( Y \) to \( XW \)) is \( 45 - 15 = 30 \)? Wait, no, \( UY = 15 \), so the height of the small triangle is 30, and the big triangle is 45. So the ratio of heights is \( \frac{30}{45}=\frac{2}{3} \). Then the ratio of the bases: \( \frac{XW}{XV}=\frac{2}{3} \). Let \( VW = x \), so \( XV = 50 + x \). Then \( \frac{50}{50 + x}=\frac{2}{3} \). Cross - multiply: \( 3\times50=2\times(50 + x)\), \( 150 = 100+2x \), \( 2x = 50 \), \( x = 25 \). Wait, but let's check again. Alternatively, the ratio of \( UY \) to \( UX \) is \( \frac{15}{45}=\frac{1}{3} \), so the ratio of the bases (since the triangles are similar) should be \( \frac{VW}{XW}=\frac{1}{2} \)? No, maybe the correct approach is: The two triangles are similar, so \( \frac{UY}{XY}=\frac{VW}{XW} \). \( UY = 15 \), \( XY = 45 - 15 = 30 \), so \( \frac{15}{30}=\frac{1}{2} \), so \( \frac{VW}{50}=\frac{1}{2} \), \( VW = 25 \). Yes, that makes sense. So the ratio of the sides of the similar triangles is \( \frac{UY}{XY}=\frac{15}{30}=\frac{1}{2} \), so the ratio of \( VW \) to \( XW \) is also \( \frac{1}{2} \), so \( VW=\frac{50}{2}=25 \).

Step2: Solve for \( VW \)

Using the similarity of triangles, the ratio of the corresponding sides (the segments on the height and the base) are equal. The ratio of \( UY \) (15) to \( XY \) (45 - 15 = 30) is \( \frac{15}{30}=\frac{1}{2} \). Since the triangles are similar, the ratio of \( VW \) to \( XW \) (50) is also \( \frac{1}{2} \). So \( VW=\frac{50}{2}=25 \).

Answer:

\( 25 \)