To find the area of the given right triangle, we can break the problem down into the following steps:
1. Identify the given values:
- Adjacent leg (base, \( b \)) \( = 27.6 \) cm
- Hypotenuse (c) \( = 30 \) cm
2. Calculate the length of the opposite leg (height, \( h \)):
- We use the Pythagorean theorem for this, \( c^2 = a^2 + b^2 \), where \( c \) is the hypotenuse, \( a \) is the adjacent side, and \( b \) is the opposite side.
- Rearrange the equation to solve for the opposite side (height, \( h \)):
[tex]\[ h^2 = c^2 - a^2 \][/tex]
[tex]\[ h = \sqrt{c^2 - a^2} \][/tex]
- Substitute the given values:
[tex]\[ h = \sqrt{30^2 - 27.6^2} \][/tex]
[tex]\[ h \approx 11.8 \][/tex] cm (approximation rounded to one decimal place for simplicity)
3. Calculate the area of the triangle using the formula \( A = \frac{1}{2} b h \):
- Substitute the values of the base \( b = 27.6 \) cm and height \( h \approx 11.8 \) cm:
[tex]\[ A = \frac{1}{2} \times 27.6 \times 11.8 \][/tex]
- Calculate the area:
[tex]\[ A \approx \frac{1}{2} \times 27.6 \times 11.8 \][/tex]
[tex]\[ A \approx 162.3 \][/tex] cm\(^2\) (approximation rounded to one decimal place)
Therefore, the approximate area of the triangle is \( 162.3 \) square centimeters.
Comparing this to the provided options:
- \( 68.7 \) cm\(^2\)
- \( 161.8 \) cm\(^2\)
- \( 381.3 \) cm\(^2\)
- \( 450.0 \) cm\(^2\)
The closest option is \( 161.8 \) cm\(^2\).
Thus, the approximate area of the triangle is [tex]\( 161.8 \)[/tex] cm[tex]\(^2\)[/tex].
