Question

question 5 of 10
what is the scale factor from δuvw to δxyz?

image of two triangles: δuvw with sides uv=3, uw=2, vw=4; δxyz with sides xz=14, xy=21, zy=28

a. 1/7
b. 7
c. 1/30
d. 30

Explanation:

Step1: Identify corresponding sides

First, we need to find the corresponding sides of the two triangles \( \triangle UVW \) and \( \triangle XYZ \). Let's assume \( VW \) corresponds to \( ZY \), \( UV \) corresponds to \( XZ \), and \( UW \) corresponds to \( XY \).

Step2: Calculate the ratio of corresponding sides

For \( VW = 4 \) and \( ZY = 28 \), the ratio is \( \frac{VW}{ZY}=\frac{4}{28}=\frac{1}{7} \). Let's check another pair. For \( UV = 3 \) and \( XZ = 14 \), wait, no, maybe I mixed up. Wait, the scale factor from \( \triangle UVW \) to \( \triangle XYZ \) means we take a side of \( \triangle UVW \) and divide by the corresponding side of \( \triangle XYZ \)? Wait, no, scale factor from A to B is (length in B)/(length in A). Wait, no, actually, scale factor from \( \triangle UVW \) to \( \triangle XYZ \) is the ratio of a side of \( \triangle XYZ \) to the corresponding side of \( \triangle UVW \). Wait, let's check the sides. \( VW = 4 \), \( ZY = 28 \). So \( 28 \div 4 = 7 \)? Wait, no, maybe I got the direction wrong. Wait, the question is "scale factor from \( \triangle UVW \) to \( \triangle XYZ \)". So that means we are going from the smaller triangle \( \triangle UVW \) to the larger triangle \( \triangle XYZ \). So the scale factor is (length in \( \triangle XYZ \)) / (length in \( \triangle UVW \)). Wait, but let's check the sides. \( VW = 4 \), \( ZY = 28 \). So \( 28 / 4 = 7 \)? But option A is \( \frac{1}{7} \), B is 7. Wait, maybe I mixed up. Wait, let's list the sides:

\( \triangle UVW \): sides \( UV = 3 \), \( UW = 2 \), \( VW = 4 \)

\( \triangle XYZ \): sides \( XZ = 14 \), \( XY = 21 \), \( ZY = 28 \)

So corresponding sides: \( UV \) corresponds to \( XZ \) (3 and 14? No, 3*7=21? Wait, no, 3*7=21, but XZ is 14. Wait, maybe \( UW = 2 \), \( XY = 21 \)? No, that doesn't match. Wait, maybe \( VW = 4 \), \( ZY = 28 \) (4*7=28). \( UV = 3 \), \( XZ = 14 \)? No, 3*7=21, not 14. Wait, maybe \( UW = 2 \), \( XY = 14 \)? 2*7=14. Ah, there we go. So \( UW = 2 \) (side of \( \triangle UVW \)) and \( XY = 14 \) (side of \( \triangle XYZ \)). So \( 14 \div 2 = 7 \)? Wait, no, scale factor from \( \triangle UVW \) to \( \triangle XYZ \) is (length in \( \triangle XYZ \)) / (length in \( \triangle UVW \)). So for \( UW = 2 \) (in \( \triangle UVW \)) and \( XY = 14 \) (in \( \triangle XYZ \)), the ratio is \( 14/2 = 7 \)? But wait, earlier with \( VW = 4 \) and \( ZY = 28 \), \( 28/4 = 7 \). And \( UV = 3 \), \( XZ = 21 \)? Wait, no, XZ is 14? Wait, the diagram: \( XZ = 14 \), \( XY = 21 \), \( ZY = 28 \). \( UV = 3 \), \( UW = 2 \), \( VW = 4 \). So corresponding sides: \( UV \) corresponds to \( XZ \)? 3 and 14? No, that's not a ratio. Wait, maybe \( UW \) corresponds to \( XZ \)? \( UW = 2 \), \( XZ = 14 \), ratio 14/2=7. \( VW = 4 \), \( ZY = 28 \), ratio 28/4=7. \( UV = 3 \), \( XY = 21 \), ratio 21/3=7. Ah, there we go. I misread \( XZ \) as 14, but maybe \( XZ \) is 21? Wait, no, the diagram: \( X \) is the top, \( Z \) and \( Y \) are the base. So \( XZ = 14 \), \( XY = 21 \), \( ZY = 28 \). \( U \) is the top, \( V \) and \( W \) are the base. \( UV = 3 \), \( UW = 2 \), \( VW = 4 \). So maybe the corresponding sides are \( UV \) to \( XY \), \( UW \) to \( XZ \), \( VW \) to \( ZY \). So \( UV = 3 \), \( XY = 21 \): \( 21/3 = 7 \). \( UW = 2 \), \( XZ = 14 \): \( 14/2 = 7 \). \( VW = 4 \), \( ZY = 28 \): \( 28/4 = 7 \). So the scale factor from \( \triangle UVW \) to \( \triangle XYZ \) is 7? Wait, but option A is \( 1/7 \). Wait, maybe I got the direction reversed. Wait, scale factor from \( \triangle UVW \) to \( \triangle XYZ \) is (length in \( \triangle XYZ \)) / (length in \( \triangle UVW \)). So if \( \triangle UVW \) is the original, and \( \triangle XYZ \) is the image, then scale factor is image length / original length. So original is \( \triangle UVW \), image is \( \triangle XYZ \). So take a side of \( \triangle UVW \), say \( VW = 4 \), corresponding side in \( \triangle XYZ \) is \( ZY = 28 \). So scale factor is \( 28 / 4 = 7 \). So the answer should be B. 7. Wait, but let's check again. If the scale factor from \( \triangle UVW \) to \( \triangle XYZ \) is 7, that means \( \triangle XYZ \) is 7 times larger than \( \triangle UVW \). Let's check the sides: 4*7=28, 2*7=14, 3*7=21. Which matches \( ZY=28 \), \( XZ=14 \), \( XY=21 \). Yes, that works. So the scale factor is 7.

Answer:

B. 7