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A right triangle has one angle that measures [tex]23^{\circ}[/tex]. The adjacent leg measures 27.6 cm, and the hypotenuse measures 30 cm.

What is the approximate area of the triangle? Round to the nearest tenth.

Area of a triangle [tex]= \frac{1}{2} \times b \times h[/tex]

A. [tex]68.7 \, \text{cm}^2[/tex]
B. [tex]161.8 \, \text{cm}^2[/tex]
C. [tex]381.3 \, \text{cm}^2[/tex]
D. [tex]450.0 \, \text{cm}^2[/tex]



Answer :

GinnyAnswer
To find the area of the right triangle, we can use the formula for the area of a triangle given by [tex]\( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \)[/tex].

1. Given Data:
- One of the angles of the right triangle is [tex]\(23^\circ\)[/tex].
- The length of the adjacent leg to this angle is 27.6 cm.
- The length of the hypotenuse is 30 cm.

2. Calculate the angle in radians:
[tex]\[ \text{Angle in radians} = 0.4014257279586958 \][/tex]

3. Calculate the sine of the angle [tex]\(23^\circ\)[/tex]:
[tex]\[ \sin(23^\circ) = 0.39073112848927377 \][/tex]

4. Find the length of the opposite leg using the sine function:
Since [tex]\(\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\)[/tex],
[tex]\[ \text{opposite leg} = \text{hypotenuse} \times \sin(23^\circ) \][/tex]
[tex]\[ \text{opposite leg} = 30 \times 0.39073112848927377 \][/tex]
[tex]\[ \text{opposite leg} = 11.721933854678213 \ \text{cm} \][/tex]

5. Calculate the area:
The area of the triangle can be calculated as:
[tex]\[ \text{Area} = \frac{1}{2} \times \text{adjacent leg} \times \text{opposite leg} \][/tex]
[tex]\[ \text{Area} = \frac{1}{2} \times 27.6 \times 11.721933854678213 \][/tex]
[tex]\[ \text{Area} = 161.76268719455933 \ \text{cm}^2 \][/tex]

6. Round the area to the nearest tenth:
[tex]\[ \text{Area} \approx 161.8 \ \text{cm}^2 \][/tex]

Therefore, the approximate area of the triangle is [tex]\( 161.8 \ \text{cm}^2 \)[/tex], which matches one of the given choices. So, the correct choice is:

[tex]\[ \boxed{161.8 \ \text{cm}^2} \][/tex]

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