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10. 在

\triangle A B C

中，如果

a=4, b=5, c=\sqrt{61}

, 那么

\angle C

的值是 ( ) A ...

May 26, 2024

Solution by Steps

step 2

Substitute the given values into the equation:

4^2 + 5^2 - 2 \cdot 4 \cdot 5 \cdot \cos(C) = (\sqrt{61})^2

step 3

Simplify the equation:

16 + 25 - 40\cos(C) = 61

step 4

Combine like terms:

41 - 40\cos(C) = 61

step 5

Solve for

\cos(C)

-40\cos(C) = 20 \implies \cos(C) = -\frac{1}{2}

step 6

Find

\angle C

\angle C = \cos^{-1}\left(-\frac{1}{2}\right) = \frac{2\pi}{3}

Key Concept

Cosine Rule

Explanation

The cosine rule relates the lengths of the sides of a triangle to the cosine of one of its angles. It is used to find an angle when all three sides of the triangle are known.

Solution by Steps

step 2

First, calculate the semi-perimeter

s

s = \frac{a+b+c}{2} = \frac{4+5+\sqrt{61}}{2} = \frac{9+\sqrt{61}}{2}

step 3

Use Heron's formula:

\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}

step 4

Substitute the values:

\text{Area} = \sqrt{\left(\frac{9+\sqrt{61}}{2}\right)\left(\frac{9+\sqrt{61}}{2} - 4\right)\left(\frac{9+\sqrt{61}}{2} - 5\right)\left(\frac{9+\sqrt{61}}{2} - \sqrt{61}\right)}

step 5

Simplify the expression:

\text{Area} = \sqrt{\left(\frac{9+\sqrt{61}}{2}\right)\left(\frac{1+\sqrt{61}}{2}\right)\left(\frac{-1+\sqrt{61}}{2}\right)\left(\frac{9-\sqrt{61}}{2}\right)}

step 6

Further simplify:

\text{Area} = \sqrt{\left(\frac{(9+\sqrt{61})(1+\sqrt{61})(-1+\sqrt{61})(9-\sqrt{61})}{16}\right)}

step 7

Calculate the area:

\text{Area} = 5\sqrt{3} \approx 8.66025

Key Concept

Heron's Formula

Explanation

Heron's formula allows us to find the area of a triangle when the lengths of all three sides are known. It involves calculating the semi-perimeter and then using it in the formula.