https://file.aiquickdraw.com/mathbot/file/17254596815695bqq7ts8.jpg

The problem shown in the image involves a right triangle with a hypotenuse \( r = 2 \times 32 \) units (likely inches or centimeters). The angle between the hypotenuse and one of the legs is \( 45^\circ \), and the lengths of the other sides \( a \), \( b \), and \( c \) need to be determined.

Here's how to approach it:

### 1. Given Information:
- \( r = 2 \times 32 = 64 \)
- \( \theta = 45^\circ \) (angle between the hypotenuse and side \( b \))
- \( a \), \( b \), and \( c \) are unknown

### 2. Relationships in a Right Triangle:
Since \( \theta = 45^\circ \), and \( \theta \) is between \( r \) and side \( b \), we can use trigonometric identities:

\[
\sin(45^\circ) = \frac{a}{r}, \quad \cos(45^\circ) = \frac{b}{r}
\]

But since \( \sin(45^\circ) = \cos(45^\circ) = \frac{1}{\sqrt{2}} \):

\[
a = \frac{r}{\sqrt{2}} = \frac{64}{\sqrt{2}} = 64 \times \frac{\sqrt{2}}{2} = 32\sqrt{2}
\]
\[
b = \frac{r}{\sqrt{2}} = \frac{64}{\sqrt{2}} = 64 \times \frac{\sqrt{2}}{2} = 32\sqrt{2}
\]

### 3. Finding Side \( c \):
Side \( c \) is given directly as 240 units in the image.

### Final Answer:
- \( a = 32\sqrt{2} \)
- \( b = 32\sqrt{2} \)
- \( c = 240 \)

Would you like more details on any part of the calculation? Here are some additional questions related to this problem:

1. What is the Pythagorean Theorem, and how does it apply to this problem?
2. How do trigonometric identities relate to the sides of a right triangle?
3. What are the special properties of a \( 45^\circ-45^\circ-90^\circ \) triangle?
4. How can you calculate \( \sqrt{2} \) in decimal form to find more precise values for \( a \) and \( b \)?
5. How does changing the angle \( \theta \) affect the lengths of \( a \) and \( b \)?

**Tip:** When working with trigonometric functions, always ensure the angle is correctly interpreted based on the given problem, as it significantly impacts the calculations.

Find the lengths of sides a, b, and c in the right triangle given r = 64 units and angle θ = 45°.

Learn how to find the side lengths of a right triangle with a 45° angle and a hypotenuse of 64 units using trigonometric identities and the Pythagorean theorem. This problem is ideal for students in Grades 9-10.

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