How to solve a triangle with 3 sides?
Understand the Problem
The question is asking for methods to solve a triangle when the lengths of all three sides are known. This refers to concepts in geometry, particularly using the triangle's sides to find angles or characteristics of the triangle using the Law of Cosines or Heron's formula.
Answer
1. Calculate semi-perimeter: \( s = \frac{a + b + c}{2} \). 2. Area: \( A = \sqrt{s(s - a)(s - b)(s - c)} \). 3. Angles: \( A = \arccos\left(\frac{b^2 + c^2 - a^2}{2bc}\right) \), \( B = \arccos\left(\frac{a^2 + c^2 - b^2}{2ac}\right) \), \( C = \arccos\left(\frac{a^2 + b^2 - c^2}{2ab}\right) \).
Answer for screen readers
To solve a triangle with known side lengths (a), (b), and (c):
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Calculate the semi-perimeter: $$ s = \frac{a + b + c}{2} $$
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Calculate the area using Heron's formula: $$ A = \sqrt{s(s - a)(s - b)(s - c)} $$
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Determine the angles using the Law of Cosines: $$ A = \arccos\left(\frac{b^2 + c^2 - a^2}{2bc}\right) $$ $$ B = \arccos\left(\frac{a^2 + c^2 - b^2}{2ac}\right) $$ $$ C = \arccos\left(\frac{a^2 + b^2 - c^2}{2ab}\right) $$
Steps to Solve
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Identify the lengths of the sides First, label the sides of the triangle as (a), (b), and (c). Make sure you know the values of each side.
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Apply Heron's formula to find the area Calculate the semi-perimeter (s) using the formula: $$ s = \frac{a + b + c}{2} $$ Then, use Heron's formula to find the area (A): $$ A = \sqrt{s(s - a)(s - b)(s - c)} $$
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Use the Law of Cosines to find angles To find the angles, you can apply the Law of Cosines. For angle (A) opposite side (a): $$ \cos A = \frac{b^2 + c^2 - a^2}{2bc} $$ You can find angle (B) and (C) similarly: $$ \cos B = \frac{a^2 + c^2 - b^2}{2ac} $$ $$ \cos C = \frac{a^2 + b^2 - c^2}{2ab} $$
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Calculate the angles Use the inverse cosine function (arccos) to find the angles in degrees: $$ A = \arccos\left(\frac{b^2 + c^2 - a^2}{2bc}\right) $$ $$ B = \arccos\left(\frac{a^2 + c^2 - b^2}{2ac}\right) $$ $$ C = \arccos\left(\frac{a^2 + b^2 - c^2}{2ab}\right) $$
To solve a triangle with known side lengths (a), (b), and (c):
-
Calculate the semi-perimeter: $$ s = \frac{a + b + c}{2} $$
-
Calculate the area using Heron's formula: $$ A = \sqrt{s(s - a)(s - b)(s - c)} $$
-
Determine the angles using the Law of Cosines: $$ A = \arccos\left(\frac{b^2 + c^2 - a^2}{2bc}\right) $$ $$ B = \arccos\left(\frac{a^2 + c^2 - b^2}{2ac}\right) $$ $$ C = \arccos\left(\frac{a^2 + b^2 - c^2}{2ab}\right) $$
More Information
When you know all three side lengths of a triangle, you can find both the area and the internal angles using Heron's formula and the Law of Cosines, respectively. This is useful in various fields, including physics, engineering, and architecture.
Tips
- Mixing up sides and angles: Ensure you correctly apply the formulas to the correct sides.
- Incorrectly using the Law of Cosines: Make sure you understand which angle corresponds to each formula.
- Forgetting to take the square root in Heron’s formula: Double-check that you’re calculating the area correctly.
