Examples of solving similar triangle problems using proportions.

You've provided two excellent examples of using similar triangles and proportions to solve real-world problems. Let's break down each one clearly.

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## 🔹 Example 1: Similar Triangles with Side Lengths

> Problem: The sides of a triangle measure 6, 11, and 15. If the smallest side of a similar triangle measures 4, find the length of its longest side.

✔ Solution:

- The original triangle has sides: 6 (smallest), 11, 15 (longest).
- The similar triangle has smallest side = 4, and we need to find its longest side, which we’ll call x.

Since the triangles are similar, their corresponding sides are in proportion.

So we set up a proportion using the smallest side and longest side from each triangle:

```
(smallest side of first) / (smallest side of second) = (longest side of first) / (longest side of second)
```

→ `6 / 4 = 15 / x`

Now solve for `x`:

Multiply both sides by `4x` to eliminate denominators:

```
6x = 4 * 15
6x = 60
x = 60 / 6
x = 10
```

✔ Answer: The longest side of the similar triangle is 10 units.

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## 🔹 Example 2: Height and Shadow (Indirect Measurement)

> Problem: On level ground, a person 6 ft tall casts a shadow of 8 ft. At the same time, a nearby tree casts a shadow of 20 ft. Find the height of the tree.

✔ Solution:

This is a classic indirect measurement problem using similar triangles.

At the same time of day, the sun’s rays hit both the person and the tree at the same angle, so the triangles formed by:
- Person + their shadow
- Tree + its shadow

are similar right triangles.

So, we can set up a proportion:

```
(height of person) / (shadow of person) = (height of tree) / (shadow of tree)
```

→ `6 / 8 = x / 20`

Now solve for `x`:

Cross-multiply:

```
6 * 20 = 8 * x
120 = 8x
x = 120 / 8
x = 15
```

✔ Answer: The height of the tree is 15 feet.

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## 🧠 Key Concept: Similar Triangles

When two triangles are similar:
- Their corresponding angles are equal
- Their corresponding sides are proportional

We use this to set up ratios and solve for unknown lengths — very useful in geometry, architecture, surveying, and everyday life!

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✔ Final Answers:
- Example 1: Longest side = 10
- Example 2: Tree height = 15 ft

Let me know if you’d like to try more problems like these!