Find the length of the remaining side of each triangle. Do not use a calculator.
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1
Identify the type of triangle given in the image (e.g., right triangle, isosceles, etc.).
If it's a right triangle, apply the Pythagorean theorem: \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse.
If it's not a right triangle, consider using the Law of Cosines: \( c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \), where \( C \) is the angle opposite side \( c \).
Substitute the known side lengths and angles into the appropriate formula.
Solve the equation for the unknown side length.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Triangle Properties
Understanding the properties of triangles is essential for solving problems related to their sides and angles. The sum of the interior angles in any triangle is always 180 degrees, and the relationship between the sides and angles is governed by the laws of sines and cosines. These properties help in determining unknown side lengths when certain angles and other side lengths are known.
Trigonometric ratios, such as sine, cosine, and tangent, relate the angles of a triangle to the lengths of its sides. For example, in a right triangle, the sine of an angle is the ratio of the length of the opposite side to the hypotenuse, while the cosine is the ratio of the adjacent side to the hypotenuse. These ratios are fundamental for calculating unknown side lengths when angles are provided.
The Law of Sines and the Law of Cosines are critical tools for solving triangles that are not right-angled. The Law of Sines states that the ratio of a side length to the sine of its opposite angle is constant for all three sides of a triangle. The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles, allowing for the calculation of unknown sides or angles when given sufficient information.