consider the two triangles. how can the triangles be proven similar by the sss similarity theorem?
show that the ratios $\frac{uv}{xy}$, $\frac{wu}{zx}$, and $\frac{wv}{zy}$ are equivalent.
show that the ratios $\frac{uv}{zy}$, $\frac{wu}{zx}$, and $\frac{wv}{xy}$ are equivalent.
show that the ratios $\frac{uv}{xy}$ and $\frac{wv}{zy}$ are equivalent, and $\angle v \cong \angle y$.
show that the ratios $\frac{uv}{zy}$ and $\frac{wu}{zx}$ are equivalent, and $\angle u \cong \angle z$.

Question

Accepted Answer

# Brief Explanations:
To prove similarity by SSS (Side - Side - Side) similarity theorem, we need to show that the ratios of the corresponding sides of the two triangles are equal.

First, let's identify the sides of the two triangles. In triangle $UVW$, the sides are $UV = 50$, $WU=40$, $WV = 60$. In triangle $ZXY$, the sides are $ZY = 48$, $ZX = 32$, $XY=40$.

Now, let's find the ratios of the corresponding sides:

- For $\frac{UV}{ZY}=\frac{50}{48}=\frac{25}{24}$
- For $\frac{WU}{ZX}=\frac{40}{32}=\frac{5}{4}=\frac{30}{24}$? Wait, no, wait. Wait, maybe I mis - identified the corresponding sides. Wait, let's re - check. Wait, $WU = 40$, $ZX=32$; $UV = 50$, $ZY = 48$; $WV=60$, $XY = 40$? No, that can't be. Wait, maybe the correct correspondence is:

Wait, let's list the sides of triangle $UVW$: $WU = 40$, $UV = 50$, $WV=60$

Sides of triangle $ZXY$: $ZX = 32$, $XY = 40$, $ZY=48$

Now, let's find the ratios:

$\frac{WU}{ZX}=\frac{40}{32}=\frac{5}{4}$

$\frac{UV}{ZY}=\frac{50}{48}=\frac{25}{24}$? No, that's not right. Wait, maybe I made a mistake. Wait, let's check the second option:

The second option says show that $\frac{UV}{ZY},\frac{WU}{ZX},\frac{WV}{XY}$ are equivalent.

Let's calculate each ratio:

$\frac{UV}{ZY}=\frac{50}{48}=\frac{25}{24}$? No, wait $UV = 50$, $ZY = 48$; $WU = 40$, $ZX = 32$; $WV=60$, $XY = 40$

Wait, $\frac{WU}{ZX}=\frac{40}{32}=\frac{5}{4}$

$\frac{UV}{ZY}=\frac{50}{48}=\frac{25}{24}$? No, that's not equal. Wait, maybe I mixed up the sides. Wait, maybe the sides of triangle $UVW$ are $WU = 40$, $WV = 60$, $UV=50$ and triangle $ZXY$ has $ZX = 32$, $XY = 40$, $ZY = 48$

Wait, $\frac{WU}{XY}=\frac{40}{40} = 1$, $\frac{WV}{ZY}=\frac{60}{48}=\frac{5}{4}$, $\frac{UV}{ZX}=\frac{50}{32}=\frac{25}{16}$. No, that's not. Wait, the second option is $\frac{UV}{ZY},\frac{WU}{ZX},\frac{WV}{XY}$

Let's recalculate:

$UV = 50$, $ZY = 48$, so $\frac{UV}{ZY}=\frac{50}{48}=\frac{25}{24}$

$WU = 40$, $ZX = 32$, so $\frac{WU}{ZX}=\frac{40}{32}=\frac{5}{4}=\frac{30}{24}$

$WV = 60$, $XY = 40$, so $\frac{WV}{XY}=\frac{60}{40}=\frac{3}{2}=\frac{36}{24}$. No, that's not equal. Wait, maybe I have the wrong correspondence.

Wait, maybe the triangles are $ \triangle UVW$ and $ \triangle ZXY$ with sides:

$UV = 50$, $WU = 40$, $WV=60$

$ZY = 48$, $ZX = 32$, $XY = 40$

Wait, let's check the second option again. The second option is $\frac{UV}{ZY},\frac{WU}{ZX},\frac{WV}{XY}$

$\frac{UV}{ZY}=\frac{50}{48}=\frac{25}{24}$

$\frac{WU}{ZX}=\frac{40}{32}=\frac{5}{4}=\frac{30}{24}$

$\frac{WV}{XY}=\frac{60}{40}=\frac{3}{2}=\frac{36}{24}$. No, that's not equal. Wait, maybe I made a mistake in the side lengths. Wait, the first triangle: $U - W$ is 40, $W - V$ is 60, $U - V$ is 50. The second triangle: $Z - X$ is 32, $X - Y$ is 40, $Z - Y$ is 48.

Wait, let's check the ratios for the second option:

$\frac{UV}{ZY}=\frac{50}{48}=\frac{25}{24}$

$\frac{WU}{ZX}=\frac{40}{32}=\frac{5}{4}=\frac{30}{24}$

$\frac{WV}{XY}=\frac{60}{40}=\frac{3}{2}=\frac{36}{24}$. No, that's not equal. Wait, maybe the correct correspondence is $ \triangle UVW$ and $ \triangle XYZ$ or something else. Wait, maybe I messed up the labels.

Wait, the SSS similarity theorem states that if the ratios of the three corresponding sides of two triangles are equal, then the triangles are similar.

Let's check the second option again. Wait, maybe the side lengths are:

Triangle 1: $WU = 40$, $UV = 50$, $WV = 60$

Triangle 2: $ZX = 32$, $XY = 40$, $ZY = 48$

Now, let's compute the ratios:

$\frac{WU}{XY}=\frac{40}{40}=1$

$\frac{UV}{ZY}=\frac{50}{48}=\frac{25}{24}$

$\frac{WV}{ZX}=\frac{60}{32}=\frac{15}{8}$. No.

Wait, the second option is $\frac{UV}{ZY},\frac{WU}{ZX},\frac{WV}{XY}$

$\frac{UV}{ZY}=\frac{50}{48}=\frac{25}{24}$

$\frac{WU}{ZX}=\frac{40}{32}=\frac{5}{4}=\frac{30}{24}$

$\frac{WV}{XY}=\frac{60}{40}=\frac{3}{2}=\frac{36}{24}$. No, that's not equal. Wait, maybe I have the wrong triangle sides. Wait, maybe the first triangle has sides $WU = 40$, $WV = 60$, $UV = 50$ and the second triangle has $ZX = 32$, $ZY = 48$, $XY = 40$

Now, let's check the ratios for the second option:

$\frac{UV}{ZY}=\frac{50}{48}=\frac{25}{24}$

$\frac{WU}{ZX}=\frac{40}{32}=\frac{5}{4}=\frac{30}{24}$

$\frac{WV}{XY}=\frac{60}{40}=\frac{3}{2}=\frac{36}{24}$. No, that's not equal. Wait, maybe the correct answer is the second option. Wait, maybe I made a calculation mistake.

Wait, $40\div32=\frac{5}{4}$, $50\div48=\frac{25}{24}$, $60\div40=\frac{3}{2}$. No, these are not equal. Wait, maybe the sides are different. Wait, maybe the first triangle: $U - W = 40$, $W - V = 60$, $U - V = 50$. The second triangle: $Z - X = 32$, $X - Y = 40$, $Z - Y = 48$. Wait, maybe the correspondence is $U - Z$, $W - X$, $V - Y$. So $WU$ corresponds to $ZX$, $UV$ corresponds to $ZY$, $WV$ corresponds to $XY$. So the ratios are $\frac{WU}{ZX}=\frac{40}{32}=\frac{5}{4}$, $\frac{UV}{ZY}=\frac{50}{48}=\frac{25}{24}$, $\frac{WV}{XY}=\frac{60}{40}=\frac{3}{2}$. No, these are not equal. Wait, maybe the problem has a typo or I'm mis - identifying the sides.

Wait, let's check the other options. The first option: $\frac{UV}{XY},\frac{WU}{ZX},\frac{WV}{ZY}$. $UV = 50$, $XY = 40$, so $\frac{50}{40}=\frac{5}{4}$; $WU = 40$, $ZX = 32$, $\frac{40}{32}=\frac{5}{4}$; $WV = 60$, $ZY = 48$, $\frac{60}{48}=\frac{5}{4}$. Oh! Wait, I think I mis - identified the corresponding sides. So $UV$ corresponds to $XY$? No, wait $UV = 50$, $XY = 40$; $WU = 40$, $ZX = 32$; $WV = 60$, $ZY = 48$. So $\frac{UV}{XY}=\frac{50}{40}=\frac{5}{4}$, $\frac{WU}{ZX}=\frac{40}{32}=\frac{5}{4}$, $\frac{WV}{ZY}=\frac{60}{48}=\frac{5}{4}$. Ah! So the first option is $\frac{UV}{XY},\frac{WU}{ZX},\frac{WV}{ZY}$ are equivalent. Wait, but the first option says $\frac{UV}{XY},\frac{WU}{ZX},\frac{WV}{ZY}$. Let's check the first option again: "Show that the ratios $\frac{UV}{XY},\frac{WU}{ZX},$ and $\frac{WV}{ZY}$ are equivalent."

Yes, because $\frac{50}{40}=\frac{5}{4}$, $\frac{40}{32}=\frac{5}{4}$, $\frac{60}{48}=\frac{5}{4}$. So the first option is correct? Wait, but earlier I thought the second option was the one, but I mis - identified the corresponding sides.

Wait, the first option: $\frac{UV}{XY}$, $UV = 50$, $XY = 40$, ratio $\frac{50}{40}=\frac{5}{4}$

$\frac{WU}{ZX}$, $WU = 40$, $ZX = 32$, ratio $\frac{40}{32}=\frac{5}{4}$

$\frac{WV}{ZY}$, $WV = 60$, $ZY = 48$, ratio $\frac{60}{48}=\frac{5}{4}$

So all three ratios are equal ($\frac{5}{4}$), so by SSS similarity theorem, the triangles are similar. So the first option is correct. Wait, but let's check the options again.

Wait, the first option is: "Show that the ratios $\frac{UV}{XY},\frac{WU}{ZX},$ and $\frac{WV}{ZY}$ are equivalent."

Yes, that's the correct way to apply SSS similarity, by showing the three corresponding side ratios are equal.

# Answer:
The correct option is: Show that the ratios $\frac{UV}{XY},\frac{WU}{ZX},$ and $\frac{WV}{ZY}$ are equivalent. (The first option)

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Answer: