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In this image, you are tasked with finding the exact values of the side lengths \( c \) and \( d \) for the given special right triangles. Let's break down the two triangles:

1. **Triangle on the left (45°-45°-90° triangle)**:
   - This is an isosceles right triangle.
   - In a 45°-45°-90° triangle, the sides are in the ratio \( 1:1:\sqrt{2} \).
   - Here, one leg is given as 2.
   - So, the hypotenuse \( c \) is calculated as:
     \[
     c = 2 \times \sqrt{2} = 2\sqrt{2}
     \]

2. **Triangle on the right (30°-60°-90° triangle)**:
   - In a 30°-60°-90° triangle, the sides are in the ratio \( 1:\sqrt{3}:2 \), where:
     - The shortest side (opposite 30°) is 5.
     - The side opposite 60° is \( d \), which is \( 5\sqrt{3} \).

So, the exact values are:
- \( c = 2\sqrt{2} \)
- \( d = 5\sqrt{3} \)

Would you like any further explanation or have any questions?

Here are 5 related questions to expand your understanding:
1. How would the ratio of sides change for a 30°-60°-90° triangle if the shortest side was doubled?
2. Can you derive the formula for the hypotenuse in a 45°-45°-90° triangle using the Pythagorean theorem?
3. What are the general properties of special right triangles that make them easy to solve?
4. How can you use special right triangles in real-world applications, such as architecture?
5. How do trigonometric functions apply to these specific types of triangles?

**Tip**: Memorizing the side ratios for special right triangles (45°-45°-90° and 30°-60°-90°) can save a lot of time in solving geometry problems.

For the right triangles below, find the exact values of the side lengths c and d.

Learn how to find the exact values of side lengths in 45°-45°-90° and 30°-60°-90° triangles. This problem involves understanding special right triangles and applying geometric formulas. Suitable for students in Grades 8-10.

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