Multiple Choice
Given a triangle where = , = , and angle = , use the Law of Sines to determine the type of triangle that can be formed.
66
views
7. Non-Right Triangles
Law of Sines
Problem 12
Textbook Question
Find the length of each side labeled a. Do not use a calculator.
<IMAGE>
Verified step by step guidance1
Identify the type of triangle and the given information. Since the problem asks for the length of side \( a \) and mentions not to use a calculator, it likely involves special right triangles or exact trigonometric ratios.
Recall the properties of special right triangles, such as the 30°-60°-90° triangle or the 45°-45°-90° triangle, where side lengths have known ratios. For example, in a 30°-60°-90° triangle, the sides are in the ratio \( 1 : \sqrt{3} : 2 \).
Use the given angle measures and the known ratios to set up an equation relating side \( a \) to the other sides. For instance, if the triangle is 30°-60°-90°, and you know the length of the hypotenuse or one leg, express \( a \) in terms of that length using the ratio.
If the triangle is not a special right triangle, apply the Law of Sines or Law of Cosines. For Law of Sines, use the formula \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). For Law of Cosines, use \( a^2 = b^2 + c^2 - 2bc \cos A \).
Solve the equation algebraically to express \( a \) in exact form without decimal approximations, using simplified radicals or fractions as appropriate.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Pythagorean Theorem
The Pythagorean theorem relates the lengths of the sides in a right triangle: the square of the hypotenuse equals the sum of the squares of the other two sides. It is essential for finding unknown side lengths when two sides are known.
Recommended video:
5:19
Solving Right Triangles with the Pythagorean Theorem
Special Right Triangles
Special right triangles, such as 45°-45°-90° and 30°-60°-90°, have side length ratios that allow side lengths to be found without a calculator. Recognizing these triangles helps simplify problems involving labeled sides.
Recommended video:
04:3945-45-90 Triangles
Basic Trigonometric Ratios
Sine, cosine, and tangent ratios relate angles to side lengths in right triangles. Understanding these ratios allows calculation of unknown sides when an angle and one side length are given.
Recommended video:
6:04Introduction to Trigonometric Functions
Related Videos
Related Practice