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Chapter 10: Problem 5

In questions \(1-10\) solve \(\triangle \mathrm{ABC}\) given \(\mathrm{AB}=69 \mathrm{~cm}, \mathrm{BC}=52 \mathrm{~cm}, \mathrm{AC}=49 \mathrm{~cm}\)

Short Answer

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Question: Given triangle ABC with side lengths AB = 69 cm, BC = 52 cm, and AC = 49 cm, find the angles A, B, and C. Answer: Use the Law of Cosines to first find angle C, opposite the longest side AB. Next, use the Law of Sines to find angle A. Lastly, use the angle sum of a triangle to find angle B. Once you have found angles A, B, and C, the solved triangle ABC can be presented with the discovered angles and given side lengths.

Step by step solution

01

Identify the longest side and its opposite angle

In this problem, AB is the longest side, so we will first find the angle opposite this side, which is angle C.
02

Use the Law of Cosines to find angle C

Using the Law of Cosines, we have: \(c^2 = a^2 + b^2 - 2ab \cdot \cos{C}\) where a = BC = 52 cm, b = AC = 49 cm, and c = AB = 69 cm. Plugging these values into the equation to find \(\cos{C}\), we have: \((69)^2 = (52)^2 + (49)^2 - 2(52)(49)\cdot \cos{C}\) Solve for \(\cos{C}\), and then use the inverse cosine function to find the angle C.
03

Use the Law of Sines to find angle A

Now that we have angle C, we can use the Law of Sines to find angle A: \(\frac{\sin{A}}{a} = \frac{\sin{C}}{c}\) Plug in the values for a and c, as well as the value of \(\sin{C}\) that you found in step 2. Solve for \(\sin{A}\), and then use the inverse sine function to find angle A.
04

Find angle B using angle sum of a triangle

Since the angles in a triangle must sum up to 180 degrees, we have: \(A + B + C = 180^\circ\) Plug in the known values for angles A and C and solve for angle B.
05

Present the solved triangle ABC

Once you have found angles A, B, and C, you can present the solution as: \(\triangle ABC\) with \(\angle A\), \(\angle B\), and \(\angle C\), and given side lengths AB, BC, and AC.

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Most popular questions from this chapter

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