Solving for Side Length in Similar Triangles

The question asks us to find the length of side YZ in triangle XYZ, given that triangle PQR is similar to triangle XYZ and their side lengths are provided.

Understanding Similar Triangles

Two triangles are said to be similar if:

• Their corresponding angles are equal.
• Their corresponding sides are in proportion.

The notation "triangle PQR is similar to triangle XYZ" ($\triangle PQR \sim \triangle XYZ$) tells us which vertices and sides correspond:

• Vertex P corresponds to Vertex X
• Vertex Q corresponds to Vertex Y
• Vertex R corresponds to Vertex Z

This correspondence implies that the sides opposite these vertices are proportional:

• Side PQ corresponds to Side XY
• Side QR corresponds to Side YZ
• Side PR corresponds to Side XZ

Setting up Proportions for Corresponding Sides

Because the triangles are similar, the ratio of corresponding sides is constant. We can write this as:

$$\frac{PQ}{XY} = \frac{QR}{YZ} = \frac{PR}{XZ}$$

Using the Given Side Lengths

We are given the following side lengths:

• PQ = 9 cm
• QR = 6 cm
• PR = 7.5 cm
• XY = 18 cm

We need to find the value of YZ. We can use the first part of the proportion:

$$\frac{PQ}{XY} = \frac{QR}{YZ}$$

Substitute the known values into this equation:

$$\frac{9 \text{ cm}}{18 \text{ cm}} = \frac{6 \text{ cm}}{YZ}$$

Solving for YZ

Now, we solve this equation for YZ. First, simplify the ratio on the left side:

$$\frac{9}{18} = \frac{1}{2}$$

So the equation becomes:

$$\frac{1}{2} = \frac{6}{YZ}$$

To find YZ, we can cross-multiply:

$$1 \times YZ = 2 \times 6$$

$$YZ = 12$$

The length of side YZ is 12 cm.

Checking the Ratios (Optional but helpful)

We found that YZ = 12 cm. Let's check the ratio $\frac{QR}{YZ}$:

$$\frac{QR}{YZ} = \frac{6}{12} = \frac{1}{2}$$

This matches the ratio $\frac{PQ}{XY} = \frac{9}{18} = \frac{1}{2}$. The proportionality holds, which confirms our calculation for YZ is correct.

Thus, the value of YZ is 12 cm.

Summary of Steps

1. Identify corresponding sides in similar triangles.
2. Set up a proportion using the ratio of corresponding sides.
3. Substitute the given side lengths into the proportion.
4. Solve the equation for the unknown side length.

Revision Table: Key Concepts in Similar Triangles

Additional Information: Applications of Similar Triangles

Similar triangles are a fundamental concept in geometry and have many real-world applications, such as:

• Indirect Measurement: Measuring heights of tall objects (buildings, trees) or distances across rivers using shadows or setting up similar triangles.
• Photography and Optics: The principles of similarity are used in lenses and how images are formed.
• Architecture and Engineering: Used in scaling drawings and blueprints, ensuring proportions are maintained.
• Cartography: Creating maps where geographic areas are scaled down proportionally.

Understanding the proportionality of corresponding sides is crucial for solving problems involving similar figures.