Question

A right triangle has a hypotenuse of length 18 and an angle of 35°, with a side opposite this angle of length 4. A second right triangle also has an angle of 35°, with a hypotenuse of length 9. Determine the length of the side opposite the 35° angle on the second triangle.

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Answer 1

To find the length of the side opposite the 35° angle in the second triangle, we can use the relationship between the lengths of the sides in a right triangle. Specifically, we use the sine function, which relates the length of the side opposite the angle to the hypotenuse.

The sine function is defined as follows for a right triangle:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]

In the first triangle, we have:
- Hypotenuse = 18
- Angle = 35°
- Opposite side = 4

We can check the sine of 35°:
\[
\sin(35°) = \frac{4}{18} \Rightarrow \sin(35°) \approx 0.5736 \quad \text{(calculated using a calculator)}
\]

Now we can find the length of the opposite side in the second triangle, where the hypotenuse is 9:

Using the sine of the angle:
\[
\sin(35°) = \frac{\text{opposite}}{9}
\]

Let \( x \) be the length of the opposite side in the second triangle:
\[
\sin(35°) = \frac{x}{9}
\]

Now we can solve for \( x \):
\[
x = 9 \cdot \sin(35°)
\]
Using the approximate value of \(\sin(35°) \approx 0.5736\):
\[
x = 9 \cdot 0.5736 \approx 5.1624
\]

Therefore, the length of the side opposite the 35° angle in the second triangle is approximately:
\[
\boxed{5.16} \text{ units}
\]