Answer :
To find the approximate area of the right triangle, we can follow these steps:
1. Understand the given values:
- Adjacent side (base): \( 27.6 \, \text{cm} \)
- Hypotenuse: \( 30 \, \text{cm} \)
- One angle (besides the right angle): \( 23^\circ \)
2. Find the length of the opposite side:
- Using the trigonometric function sine, we know that \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \).
- Here, \( \theta = 23^\circ \).
Thus,
[tex]\[ \sin(23^\circ) = \frac{\text{opposite}}{30} \][/tex]
Solving for the opposite side:
[tex]\[ \text{opposite} = 30 \times \sin(23^\circ) \][/tex]
Using the result:
[tex]\[ \text{opposite} \approx 11.72 \, \text{cm} \][/tex]
3. Calculate the area of the triangle:
- The formula for the area of a triangle is:
[tex]\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \][/tex]
- In this case, the base is the adjacent side, \( 27.6 \, \text{cm} \), and the height is the opposite side we calculated, \( 11.72 \, \text{cm} \).
Substituting the values:
[tex]\[ \text{Area} = \frac{1}{2} \times 27.6 \times 11.72 \][/tex]
Performing the multiplication:
[tex]\[ \text{Area} \approx 161.76 \, \text{cm}^2 \][/tex]
4. Round to the nearest tenth:
[tex]\[ \text{Area} \approx 161.8 \, \text{cm}^2 \][/tex]
Therefore, the approximate area of the triangle is [tex]\( \boxed{161.8 \, \text{cm}^2} \)[/tex]. This matches the provided result and confirms our calculations. Hence, the correct answer is [tex]\( 161.8 \, \text{cm}^2 \)[/tex].
1. Understand the given values:
- Adjacent side (base): \( 27.6 \, \text{cm} \)
- Hypotenuse: \( 30 \, \text{cm} \)
- One angle (besides the right angle): \( 23^\circ \)
2. Find the length of the opposite side:
- Using the trigonometric function sine, we know that \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \).
- Here, \( \theta = 23^\circ \).
Thus,
[tex]\[ \sin(23^\circ) = \frac{\text{opposite}}{30} \][/tex]
Solving for the opposite side:
[tex]\[ \text{opposite} = 30 \times \sin(23^\circ) \][/tex]
Using the result:
[tex]\[ \text{opposite} \approx 11.72 \, \text{cm} \][/tex]
3. Calculate the area of the triangle:
- The formula for the area of a triangle is:
[tex]\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \][/tex]
- In this case, the base is the adjacent side, \( 27.6 \, \text{cm} \), and the height is the opposite side we calculated, \( 11.72 \, \text{cm} \).
Substituting the values:
[tex]\[ \text{Area} = \frac{1}{2} \times 27.6 \times 11.72 \][/tex]
Performing the multiplication:
[tex]\[ \text{Area} \approx 161.76 \, \text{cm}^2 \][/tex]
4. Round to the nearest tenth:
[tex]\[ \text{Area} \approx 161.8 \, \text{cm}^2 \][/tex]
Therefore, the approximate area of the triangle is [tex]\( \boxed{161.8 \, \text{cm}^2} \)[/tex]. This matches the provided result and confirms our calculations. Hence, the correct answer is [tex]\( 161.8 \, \text{cm}^2 \)[/tex].
