To solve the problem of finding the area of the right triangle, we need to use the given angle and the base length. Here's a step-by-step breakdown of the solution:
1. Identify the given values:
- One angle of the right triangle is [tex]\(23^\circ\)[/tex].
- The length of the base [tex]\( b \)[/tex] is 30 cm.
2. Determine the height using trigonometric properties:
- Since the angle given is [tex]\(23^\circ\)[/tex] and it is the angle adjacent to the base, we use the tangent function. The tangent of an angle in a right triangle is the ratio of the opposite side (which will be the height [tex]\( h \)[/tex]) to the adjacent side (which is the base [tex]\( b \)[/tex]).
- Therefore, [tex]\(\tan(23^\circ) = \frac{h}{b}\)[/tex].
3. Calculate the height:
- Rearrange the equation to solve for [tex]\( h \)[/tex]: [tex]\( h = b \times \tan(23^\circ) \)[/tex].
- Substitute [tex]\( b = 30 \)[/tex] cm into the equation: [tex]\( h = 30 \cdot \tan(23^\circ) \)[/tex].
- Using the tangent value: [tex]\( \tan(23^\circ) \approx 0.424 \)[/tex].
- Thus, [tex]\( h = 30 \times 0.424 \approx 12.734244486288143 \)[/tex] cm.
4. Calculate the area of the triangle:
- The formula for the area of a triangle is [tex]\( \text{Area} = \frac{1}{2} \cdot b \cdot h \)[/tex].
- Substitute the values: [tex]\( \text{Area} = \frac{1}{2} \cdot 30 \cdot 12.734244486288143 \)[/tex].
- Perform the calculation: [tex]\( \text{Area} = 191.01366729432215 \)[/tex] cm[tex]\(^2\)[/tex].
5. Approximate and choose the closest value:
- The calculated area is approximately [tex]\( 191.01 \)[/tex] cm[tex]\(^2\)[/tex].
Comparing this to the given choices:
- [tex]\( 68.7 \)[/tex] cm[tex]\(^2\)[/tex]
- [tex]\( 161.8 \)[/tex] cm[tex]\(^2\)[/tex]
- [tex]\( 381.3 \)[/tex] cm[tex]\(^2\)[/tex]
- [tex]\( 450.0 \)[/tex] cm[tex]\(^2\)[/tex]
The closest value to [tex]\( 191.01 \)[/tex] cm[tex]\(^2\)[/tex] is [tex]\( 161.8 \)[/tex] cm[tex]\(^2\)[/tex].
Therefore, the approximate area of the triangle is [tex]\( 161.8 \)[/tex] cm[tex]\(^2\)[/tex].
