Two right angle triangles are combined to form a rectangle of length 4 cm. one triangle is isosceles. Find the diagonal length of rectangle.

Options

A. Cannot be determined
B. `16 sqrt2 cm`
C. `4 sqrt2 cm`
D. `4cm`

Answer

To solve the problem, we need to find the diagonal length of a rectangle formed by combining two right-angled triangles, one of which is isosceles. Let's break down the solution step by step.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have a rectangle formed by two right-angled triangles.
- The length of the rectangle is given as 4 cm.
- One of the triangles is isosceles.

2. **Identifying the Dimensions**:
- Since the rectangle is formed by two right-angled triangles, and one triangle is isosceles, we can assume that the two legs of the isosceles triangle are equal.
- Let’s denote the legs of the isosceles triangle as \( a \). Given that the length of the rectangle is 4 cm, we can assume that both legs of the isosceles triangle are also 4 cm.

3. **Finding the Diagonal Using Pythagorean Theorem**:
- The diagonal of the rectangle can be found using the Pythagorean theorem. In a rectangle, the diagonal forms a right triangle with the length and width of the rectangle.
- Here, both the length and width of the rectangle are equal to 4 cm (since the rectangle is formed by two isosceles triangles).
- Therefore, we can denote the diagonal as \( d \).

4. **Applying the Pythagorean Theorem**:
- According to the Pythagorean theorem:
\[ d^2 = \text{length}^2 + \text{width}^2 \]
- Substituting the values:
\[ d^2 = 4^2 + 4^2 \]
\[ d^2 = 16 + 16 \]
\[ d^2 = 32 \]

5. **Calculating the Diagonal**:
- To find \( d \), we take the square root of both sides:
\[ d = \sqrt{32} \]
- We can simplify \( \sqrt{32} \):
\[ d = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2} \text{ cm} \]

### Final Answer:
The diagonal length of the rectangle is \( 4\sqrt{2} \) cm.

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