The triangles given in the image are similar, as indicated by $\triangle ABC \sim \triangle XYZ$. This means that the corresponding sides of the triangles are proportional, and we can use this similarity to find the missing lengths of triangle $ABC$.

Step 1: Calculate the missing side of $\triangle XYZ$

In $\triangle XYZ$, two sides are given: $XY = 30$ and $YZ = 40$. To find the third side $XZ$, we can use the Pythagorean theorem:

$XZ^2 = XY^2 + YZ^2$ $XZ^2 = 30^2 + 40^2$ $XZ^2 = 900 + 1600 = 2500$ $XZ = \sqrt{2500} = 50$

So, $XZ = 50$.

Step 2: Use the similarity ratio

Since $\triangle ABC \sim \triangle XYZ$, the ratio of corresponding sides must be the same. The sides correspond as follows:

• $AB$ corresponds to $XY$,
• $BC$ corresponds to $YZ$,
• $AC$ corresponds to $XZ$.

The ratio of the corresponding sides can be expressed as: $\frac{AB}{XY} = \frac{BC}{YZ} = \frac{AC}{XZ}$

We know that $BC = 50$, $YZ = 40$, so the ratio is: $\frac{BC}{YZ} = \frac{50}{40} = 1.25$

Thus, the ratio of similarity is $1.25$.

Step 3: Find the missing sides $AB$ and $AC$

Now, using the ratio of $1.25$, we can find $AB$ and $AC$:

• To find $AB$, we use the corresponding side $XY = 30$: $AB = XY \times 1.25 = 30 \times 1.25 = 37.5$

• To find $AC$, we use the corresponding side $XZ = 50$: $AC = XZ \times 1.25 = 50 \times 1.25 = 62.5$

Final Results:

• $AB = 37.5$
• $AC = 62.5$

These are the missing lengths of $\triangle ABC$.

Would you like further details or clarification?

Here are some related questions:

1. How do we confirm that two triangles are similar?
2. How is the Pythagorean theorem used in non-right triangles?
3. What are other methods to find missing sides in triangles besides trigonometric ratios?
4. How can the proportionality constant between similar triangles be derived?
5. How do trigonometric ratios (sine, cosine, tangent) relate to this problem?

Tip: Always check for similar triangles when working with proportional side lengths to simplify your calculations!