To find the area of a triangle given the side lengths, we can use Heron's formula, which is suitable when we have all three sides.
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First, we need to find the semi-perimeter (s): \[ s = \frac{a + b + c}{2} \] where \( a = 12 , \text{cm}, b = 15 , \text{cm}, c = 10 , \text{cm} \): \[ s = \frac{12 + 15 + 10}{2} = \frac{37}{2} = 18.5 , \text{cm} \]
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Now we can apply Heron's formula for the area \( A \): \[ A = \sqrt{s(s - a)(s - b)(s - c)} \] \[ A = \sqrt{18.5(18.5 - 12)(18.5 - 15)(18.5 - 10)} \] \[ A = \sqrt{18.5(6.5)(3.5)(8.5)} \]
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Now, calculate the values: \[ 18.5 - 12 = 6.5, \quad 18.5 - 15 = 3.5, \quad 18.5 - 10 = 8.5 \] So, we compute: \[ A = \sqrt{18.5 \times 6.5 \times 3.5 \times 8.5} \]
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Evaluating \( 18.5 \times 6.5 \): \[ 18.5 \times 6.5 = 120.25 \]
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Next, \( 3.5 \times 8.5 \): \[ 3.5 \times 8.5 = 29.75 \]
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Now, multiply the results: \[ 120.25 \times 29.75 \approx 3580.1875 \]
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Finally, take the square root: \[ A \approx \sqrt{3580.1875} \approx 59.9 \]
Thus, the area of the triangle is approximately \( 59.9 , \text{cm}^2 \).
To finalize: \[ A \approx 59.9 , \text{cm}^2 \]