Question

Find the area of a triangle with sides 7 and 8 and the included angle is 45 degrees.

Answers

Answered by GPT-4o mini
To find the area of a triangle when two sides and the included angle are known, you can use the formula:

\[
\text{Area} = \frac{1}{2}ab \sin(C)
\]

where \( a \) and \( b \) are the lengths of the two sides, and \( C \) is the measure of the included angle.

In this case, we have:
- \( a = 7 \)
- \( b = 8 \)
- \( C = 45^\circ \)

Now, we can plug these values into the formula:

\[
\text{Area} = \frac{1}{2} \times 7 \times 8 \times \sin(45^\circ)
\]

We know that \( \sin(45^\circ) = \frac{\sqrt{2}}{2} \). Thus, we can substitute this value into the formula:

\[
\text{Area} = \frac{1}{2} \times 7 \times 8 \times \frac{\sqrt{2}}{2}
\]

Calculating \( \frac{1}{2} \times 7 \times 8 \):

\[
\frac{1}{2} \times 7 \times 8 = \frac{56}{2} = 28
\]

Now multiplying by \( \frac{\sqrt{2}}{2} \):

\[
\text{Area} = 28 \times \frac{\sqrt{2}}{2} = 14\sqrt{2}
\]

Thus, the area of the triangle is:

\[
\boxed{14\sqrt{2}}
\]
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