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To solve for the variables \(a\), \(b\), \(c\), and \(d\), we can use trigonometric properties of special right triangles since this diagram involves angles of 45°, 30°, and 90°. Here's how we can approach it step by step:

1. **Step 1: Understand the triangle structure**
   - The triangle consists of two special right triangles:
     - A 45°-45°-90° triangle on the left.
     - A 30°-60°-90° triangle on the right.

2. **Step 2: 45°-45°-90° Triangle Properties**
   - In a 45°-45°-90° triangle, the sides are in the ratio \(1:1:\sqrt{2}\).
   - The hypotenuse in this case is given as \(10\sqrt{2}\), so the legs of the triangle can be calculated:
     \[
     a = \frac{10\sqrt{2}}{\sqrt{2}} = 10
     \]
     So, \(a = 10\).

3. **Step 3: 30°-60°-90° Triangle Properties**
   - In a 30°-60°-90° triangle, the sides are in the ratio \(1:\sqrt{3}:2\).
   - The shorter leg \(d\) is opposite the 30° angle, and the longer leg \(c\) is opposite the 60° angle. The hypotenuse \(b\) is twice the shorter leg.
   - From the given 45°-45°-90° triangle, the side \(a = 10\) also serves as the shorter leg (\(d\)) in the 30°-60°-90° triangle.

4. **Step 4: Solve for \(b\) and \(c\)**
   - The hypotenuse \(b\) in the 30°-60°-90° triangle is twice the shorter leg \(d\), so:
     \[
     b = 2d = 2 \times 10 = 20
     \]
   - The longer leg \(c\) is \(d\sqrt{3}\):
     \[
     c = 10\sqrt{3}
     \]

Thus, the values of the variables are:

- \(a = 10\)
- \(b = 20\)
- \(c = 10\sqrt{3}\)
- \(d = 10\)

Would you like further clarification or details on any specific part of this solution?

Here are 5 related questions to explore:
1. How do the properties of a 30°-60°-90° triangle change with different hypotenuse lengths?
2. Why are the side ratios in special right triangles (45°-45°-90° and 30°-60°-90°) fixed?
3. Can the Pythagorean theorem be used to verify these results for both triangles?
4. How would the solution change if the given hypotenuse was different, say \(15\sqrt{2}\)?
5. How can these properties be applied to real-world problems involving right triangles?

**Tip:** In geometry, recognizing special triangles like 45°-45°-90° and 30°-60°-90° can save time by using known side ratios rather than calculating each side from scratch!

Find the value of each variable in a right triangle involving 45°-45°-90° and 30°-60°-90° special triangles, with a hypotenuse of 10√2.

Learn how to find the values of variables in a diagram combining a 45°-45°-90° triangle and a 30°-60°-90° triangle, given a hypotenuse of 10√2. This problem covers the application of special right triangle properties, suitable for students in Grades 9-12.

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