The triangle sides are given as follows: a = 21 (opposite to angle Q), b = 24 (opposite to angle R), c = 25 (opposite to angle S). Find the largest angle, which is opposite the lon... The triangle sides are given as follows: a = 21 (opposite to angle Q), b = 24 (opposite to angle R), c = 25 (opposite to angle S). Find the largest angle, which is opposite the longest side, angle S, using the fact that the interior angles of a triangle sum to 180 degrees.
Understand the Problem
The question is asking to find the largest angle of a triangle based on the lengths of its sides by applying the triangle angle sum property, which states that the sum of angles in a triangle equals 180 degrees.
Answer
$C = \cos^{-1}\left(\frac{a^2 + b^2 - c^2}{2ab}\right)$
Answer for screen readers
The largest angle $C$ can be found by using the law of cosines in the triangle.
Steps to Solve
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Identify the triangle sides First, denote the lengths of the sides of the triangle as $a$, $b$, and $c$, where $c$ is the longest side.
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Use the law of cosines To find the largest angle, we can apply the law of cosines. For a triangle with sides $a$, $b$, and $c$, the formula is:
$$ \cos(C) = \frac{a^2 + b^2 - c^2}{2ab} $$
Here, $C$ is the angle opposite side $c$.
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Calculate the cosine of the angle Plug in the values of $a$, $b$, and $c$ into the equation to calculate $\cos(C)$, which will help in determining the angle.
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Find the angle using inverse cosine To find the angle $C$, take the inverse cosine of the result you obtained:
$$ C = \cos^{-1}\left(\cos(C)\right) $$
- Verify the angle sum property Ensure that $C$ is indeed the largest angle by checking the other angles if needed, ensuring that the sum of all angles equals 180 degrees.
The largest angle $C$ can be found by using the law of cosines in the triangle.
More Information
Using the law of cosines allows you to find the angles of a triangle when you know the lengths of the sides. This approach is helpful in various applications, including engineering and physics.
Tips
- Forgetting to identify the longest side correctly, which can result in finding the wrong angle.
- Not using the correct formula from the law of cosines, leading to calculation errors.
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