IDNLearn.com: Your trusted source for accurate and reliable answers. Get prompt and accurate answers to your questions from our experts who are always ready to help.

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]


Sagot :

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]
Thank you for using this platform to share and learn. Don't hesitate to keep asking and answering. We value every contribution you make. Your questions deserve accurate answers. Thank you for visiting IDNLearn.com, and see you again for more solutions.