Question

Find the surface area of the figure below. (Triangle: 10cm, 9cm, 9cm)
A. 261 cm^{2}
B. 270 cm^{2}
C. 441 cm^{2}
D. 810 cm^{2}

Answer 1

To calculate the surface area of the triangle with sides measuring 10 cm, 9 cm, and 9 cm, we can use Heron's formula.

1. **Calculate the semi-perimeter (s)** of the triangle:
\[
s = \frac{a + b + c}{2}
\]
where \( a = 10 \, \text{cm} \), \( b = 9 \, \text{cm} \), \( c = 9 \, \text{cm} \).

\[
s = \frac{10 + 9 + 9}{2} = \frac{28}{2} = 14 \, \text{cm}
\]

2. **Apply Heron's formula** to find the area (A):
\[
A = \sqrt{s(s-a)(s-b)(s-c)}
\]
\[
A = \sqrt{14(14 - 10)(14 - 9)(14 - 9)}
\]

Calculating each term:
\[
14 - 10 = 4
\]
\[
14 - 9 = 5
\]

Now substitute:
\[
A = \sqrt{14 \cdot 4 \cdot 5 \cdot 5}
\]
\[
A = \sqrt{14 \cdot 100} = \sqrt{1400}
\]
\[
A = \sqrt{100 \cdot 14} = 10\sqrt{14}
\]

To find the numerical value of \( 10\sqrt{14} \), we can approximate \( \sqrt{14} \):
\[
\sqrt{14} \approx 3.74
\]
\[
10\sqrt{14} \approx 10 \cdot 3.74 = 37.4 \, \text{cm}^2
\]

However, there is a mistake; the question is asking for surface area and the given options are likely in square centimeters. Thus, it appears we need to adjust the interpretation for answers relating to total surfaces of a 3D figure (like a triangular prism or something similar) which needs clarification since the triangle dimensions alone yield a planar area.

If we consider this a geometric surface area question, then it would depend on provided height or depth for 3D shape, or if still a flat triangle area:

Since the area of the triangle calculated is **approximately 37.4 cm² (not listed)**, it may not correlate with available surface area options, suggesting more context is needed to understand other spatial relationships or if this is a prism, cylinder etc. Thus surface area would be obtained in respective volumetric definitions or mesh multiplication, without height dedicated by specification given here.

For the immediate task of identifying accurate choices based on single, flattened triangular area only which seemed asked, we don’t place direct equivalency towards 261, 270, 441 or 810 that request cumulatively spans other wrought constructs outlined not fully answered presently. Direct referential might call for clarifying base triangle augmentation or other dimensionality that bounds within full rendered outlines.

Conclusively for raw triangle field math from Heron's presented, it's suited to reflect around 37.4 so validity checks & assumptions should adjust or yield clarity based on further particulars on sought surface categorization moving forward based premises laid.