Problem 22 - Algebra 2

Question

In \(\triangle A B C, \angle C\) is a right angle. Find the remaining sides and angles. Round your answers to the nearest tenth. \(b=4.3, c=9.1\)

Step-by-Step Solution

Answer

With given \(b = 4.3\) and \(c = 9.1\) of a right triangle, side a and angles A and B can be calculated using Pythagorean theorem and trigonometric functions. The final answers would depend on the calculations at each step and need to be rounded at each respective step to the nearest tenth as required.

Step 1: Calculate for side a

Use the Pythagorean theorem \(c^2 = a^2 + b^2\) to solve for side a. Rearrange it to get \(a^2 = c^2 - b^2\). Substitute the given \(b = 4.3\) and \(c = 9.1\) into it, and then calculate the square root of the solution to get a.

Step 2: Calculate for \(\angle A\)

Use the tangent function to solve for \(\angle A\). In a right triangle, \(\tan(A) = \frac{b}{a}\). Rearrange it to get \(A = \tan^{-1}(\frac{b}{a})\). Substitute the calculated a from step 1 and given \(b = 4.3\) into it, and then calculate the inverse tangent of the solution to get angle A (in degrees).

Step 3: Calculate for \(\angle B\)

Use the sum of triangle angles which should add up to 180 degrees. Therefore, \(\angle B = 180° - (90° + \angle A)\). Substitute the calculated \(\angle A\) from step 2 into it and calculate to get \(\angle B\).

Key Concepts

Pythagorean Theorem

One of the foundational principles for solving right triangle problems is the Pythagorean Theorem. This theorem is expressed with the formula:

• \( c^2 = a^2 + b^2 \)

where \(c\) is the hypotenuse, the side opposite the right angle. The sides \(a\) and \(b\) are the other two legs of the triangle.

In our given problem, to determine the unknown side \(a\), you rearrange the formula to:

• \(a^2 = c^2 - b^2\)

By substituting in the given values, \(b = 4.3\) and \(c = 9.1\), you end up calculating the square root of \(9.1^2 - 4.3^2\).

This method allows you to find any missing side of a right triangle when you have the lengths of the other two sides. It's a fundamental concept you should feel comfortable with, as it frequently appears in geometry.

Trigonometry

Understanding trigonometry is essential for exploring relationships between the angles and sides of triangles, particularly right triangles. In this context, trigonometry offers functions like sine, cosine, and tangent to relate these dimensions.

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  The tangent function in particular, is useful when you are trying to find an angle when two sides are known.

In our specific problem, by using the tangent function, we can find \( \angle A \) in triangle \( \triangle ABC \). Trigonometry allows us to bridge the gap between side lengths and angle measures, expanding our toolkit for solving right triangles.

Tangent Function

The tangent function is particularly useful in problems involving right triangles. It is defined as the ratio of the length of the opposite side to the length of the adjacent side. For an angle \(A\) in a right triangle, this can be expressed as:

• \(\tan(A) = \frac{b}{a}\)

To solve for \(\angle A\), you use the inverse tangent function, rearranging the formula to find:

• \(A = \tan^{-1}\left(\frac{4.3}{a}\right)\)

After calculating \(a\) using the Pythagorean Theorem, plug it into this equation. Then you find \(\angle A\) using a calculator set to degrees mode.

This process showcases how tangent connects angles and side lengths, a critical tool in trigonometry.

Triangle Angle Sum

The Triangle Angle Sum theorem tells us that the sum of the angles in any triangle must add up to 180 degrees. This simple but powerful concept helps find unknown angles when the others are known.

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  In a right triangle, one of those angles is always 90 degrees, simplifying calculations.

In the problem, once \(\angle A\) is determined using the tangent function, you apply the Triangle Angle Sum theorem:

• \(\angle B = 180° - (90° + \angle A)\)

This step ensures that all angles are correctly calculated, adhering to this fundamental geometric principle. Understanding this theorem helps solve countless problems in geometry involving any type of triangle.