Similar Triangles - GCSE Maths - Free Printable

Problem Analysis:

The problem involves two similar triangles. Similar triangles have corresponding angles that are equal, and their corresponding sides are proportional. We are tasked with finding the unknown lengths \( x \) and \( y \) in both triangles.

Step-by-Step Solution:

#### Step 1: Identify Corresponding Sides
From the given diagram, the two triangles are similar. The corresponding sides are:
- In the left triangle: \( 10 \, \text{cm} \), \( 8 \, \text{cm} \), \( 13 \, \text{cm} \), \( 20 \, \text{cm} \), \( x \, \text{cm} \), and \( y \, \text{cm} \).
- In the right triangle: \( 18 \, \text{cm} \), \( 12 \, \text{cm} \), \( 15 \, \text{cm} \), \( 25 \, \text{cm} \), \( x \, \text{cm} \), and \( y \, \text{cm} \).

The side lengths \( x \) and \( y \) are the same in both triangles because they correspond to each other.

#### Step 2: Use Proportionality of Corresponding Sides
Since the triangles are similar, the ratios of corresponding sides are equal. We can set up proportions using the given side lengths.

##### For the Left Triangle:
The base of the left triangle is \( y \, \text{cm} \), and the height is \( x \, \text{cm} \). The smaller triangle within it has a base of \( 20 \, \text{cm} \) and a height of \( 13 \, \text{cm} \).

The ratio of the heights is:
\[
\frac{x}{13} = \frac{y}{20}
\]

The ratio of the bases is:
\[
\frac{y}{20} = \frac{10}{8}
\]

Simplify the ratio \( \frac{10}{8} \):
\[
\frac{10}{8} = \frac{5}{4}
\]

So, we have:
\[
\frac{y}{20} = \frac{5}{4}
\]

Solve for \( y \):
\[
y = 20 \times \frac{5}{4} = 25 \, \text{cm}
\]

Now, substitute \( y = 25 \) into the height ratio:
\[
\frac{x}{13} = \frac{25}{20}
\]

Simplify \( \frac{25}{20} \):
\[
\frac{25}{20} = \frac{5}{4}
\]

So, we have:
\[
\frac{x}{13} = \frac{5}{4}
\]

Solve for \( x \):
\[
x = 13 \times \frac{5}{4} = \frac{65}{4} = 16.25 \, \text{cm}
\]

#### Step 3: Verify with the Right Triangle
In the right triangle, the base is \( 25 \, \text{cm} \), and the height is \( x \, \text{cm} \). The smaller triangle within it has a base of \( 15 \, \text{cm} \) and a height of \( y \, \text{cm} \).

The ratio of the bases is:
\[
\frac{25}{15} = \frac{5}{3}
\]

The ratio of the heights is:
\[
\frac{x}{y} = \frac{18}{12} = \frac{3}{2}
\]

We already found \( y = 25 \, \text{cm} \) and \( x = 16.25 \, \text{cm} \). Let's verify:
\[
\frac{x}{y} = \frac{16.25}{25} = \frac{65/4}{25} = \frac{65}{100} = \frac{13}{20}
\]

This matches the proportionality, confirming our solution.

Final Answer:

\[
\boxed{x = 16.25 \, \text{cm}, y = 25 \, \text{cm}}
\]