To find the area of the triangle, let's follow a step-by-step process.
### Step 1: Identify the Known Values
- Adjacent leg (base, [tex]\( b \)[/tex]): 27.6 cm
- Hypotenuse ([tex]\( h \)[/tex]): 30 cm
### Step 2: Use the Pythagorean Theorem to Find the Opposite Leg
In a right triangle, we can use the Pythagorean Theorem [tex]\( a^2 + b^2 = h^2 \)[/tex] to find the length of the opposite leg ([tex]\( a \)[/tex]):
[tex]\[ a^2 = h^2 - b^2 \][/tex]
[tex]\[ a^2 = 30^2 - 27.6^2 \][/tex]
[tex]\[ a^2 = 900 - 761.76 \][/tex]
[tex]\[ a^2 = 138.24 \][/tex]
[tex]\[ a = \sqrt{138.24} \][/tex]
[tex]\[ a \approx 11.75755076535925 \][/tex]
### Step 3: Calculate the Area of the Triangle
The area [tex]\( A \)[/tex] of a right triangle is given by the formula:
[tex]\[ A = \frac{1}{2} \times \text{base} \times \text{height} \][/tex]
In this context, the base ([tex]\( b \)[/tex]) is the adjacent leg, and the height ([tex]\( h \)[/tex]) is the length of the opposite leg [tex]\( a \)[/tex]:
[tex]\[ A = \frac{1}{2} \times 27.6 \times 11.75755076535925 \][/tex]
[tex]\[ A \approx 162.3 \, \text{cm}^2 \][/tex]
So, the approximate area of the triangle is [tex]\( 162.3 \, \text{cm}^2 \)[/tex], rounded to the nearest tenth.
Therefore, the correct answer is [tex]\( 162.3 \, \text{cm}^2 \)[/tex].
Among the given options, none match the precisely rounded answer of [tex]\( 162.3 \, \text{cm}^2 \)[/tex]. Therefore, it seems there might be an error in the available answer choices. The correct calculations, as detailed, show that the closest answer is [tex]\( 161.8 \, \text{cm}^2 \)[/tex].
