a right triangle has a hypotenuse length of 12 cm and a 22* angle. the side opposite the 22* angle has a length of 8 cm. a second right triangle also has an angle of 22*, with a hypotenuse length of 6 cm. determine the length of the side opposite the 22* angle on the second triangle.

a. the side opposite the second triangle's 22* angle has a length of 4 cm
b. the side opposite the second triangle's 22* angle has a length of 9 cm
c. the side opposite the second triangle's 22* angle has a length of 12 cm
d. the side opposite the seccond triangle's 22* angle has a length of 16 cm

1 answer

To determine the length of the side opposite the 22° angle in the second right triangle, we can use the sine function.

In a right triangle, the sine of an angle is given by the ratio of the length of the side opposite the angle to the length of the hypotenuse:

\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]

We know the angle \( \theta = 22^\circ \) and the hypotenuse of the second triangle is 6 cm. Thus, we can express the length of the side opposite the 22° angle as follows:

\[ \text{opposite} = \sin(22^\circ) \times \text{hypotenuse} \]

Substituting in the known values:

\[ \text{opposite} = \sin(22^\circ) \times 6 \]

Using a calculator, we find that:

\[ \sin(22^\circ) \approx 0.3746 \]

Now we can calculate:

\[ \text{opposite} \approx 0.3746 \times 6 \approx 2.2476 , \text{cm} \]

Rounding this to two decimal places gives approximately 2.25 cm.

Since none of the answer choices provided (a. 4 cm, b. 9 cm, c. 12 cm, d. 16 cm) match this calculated length, it seems there might be a misunderstanding or error in the choices or the problem statement.

However, the important result is that the length of the side opposite the 22° angle in the second triangle, calculated using the hypotenuse of 6 cm, is approximately 2.25 cm.