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Chapter 12: Problem 13

Triangle Find the measures of the angles of the triangle whose vertices are \(A=(-1,0), B=(2,1),\) and \(C=(1,-2) .\)

Short Answer

Expert verified
The angles of the triangle are approximately 71.57掳, 71.57掳, and 36.86掳.

Step by step solution

01

Find the Length of Each Side

To determine the measures of the angles in the triangle, we first need to calculate the lengths of each side using the distance formula. For any two points \((x_1, y_1)\) and \((x_2, y_2)\), the distance is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Calculating for side AB: \[ AB = \sqrt{(2 - (-1))^2 + (1 - 0)^2} = \sqrt{3^2 + 1^2} = \sqrt{10} \]Calculating for side BC: \[ BC = \sqrt{(1 - 2)^2 + (-2 - 1)^2} = \sqrt{(-1)^2 + (-3)^2} = \sqrt{10} \]Calculating for side CA: \[ CA = \sqrt{(-1 - 1)^2 + (0 + 2)^2} = \sqrt{(-2)^2 + 2^2} = \sqrt{8} \].Thus, the sides are: \( AB = \sqrt{10}, BC = \sqrt{10}, CA = \sqrt{8} \).
02

Use the Cosine Rule to Find the Angles

Now that we have the lengths of the sides, we can use the cosine rule to find the angles. The cosine rule states: \( \cos(C) = \frac{a^2 + b^2 - c^2}{2ab} \), where \(a\), \(b\), and \(c\) are the lengths of the sides opposite the angles \(A\), \(B\), and \(C\) respectively.Let's find angle \(A\):\[ \cos(A) = \frac{BC^2 + CA^2 - AB^2}{2 \times BC \times CA} = \frac{10 + 8 - 10}{2 \times \sqrt{10} \times \sqrt{8}} = \frac{8}{2 \times \sqrt{80}} = \frac{1}{\sqrt{10}} \]Angle \(A = \cos^{-1}\left(\frac{1}{\sqrt{10}}\right).\) Similarlly, find angles \(B\) and \(C\):
03

Calculate \( B \) and \( C \) Using the Cosine Rule

Calculate the cosine of angle \(B \):\[ \cos(B) = \frac{CA^2 + AB^2 - BC^2}{2 \times CA \times AB} = \frac{8 + 10 - 10}{2 \times \sqrt{8} \times \sqrt{10}} = \frac{8}{2\sqrt{80}} = \frac{1}{\sqrt{10}} \] Angle \(B = \cos^{-1}\left(\frac{1}{\sqrt{10}}\right).\)Calculate angle \(C \):\[ \cos(C) = \frac{AB^2 + BC^2 - CA^2}{2 \times AB \times BC} = \frac{10 + 10 - 8}{2 \times \sqrt{10} \times \sqrt{10}} = \frac{12}{20} = \frac{3}{5} \] Angle \(C = \cos^{-1}\left(\frac{3}{5}\right).\)
04

Calculate the Measures of the Angles

Using a calculator to find the angles:Angle \(A\): \( \cos^{-1}\left(\frac{1}{\sqrt{10}}\right) \approx 71.57^\circ \)Angle \(B\): \( \cos^{-1}\left(\frac{1}{\sqrt{10}}\right) \approx 71.57^\circ \)Angle \(C\): \( \cos^{-1}\left(\frac{3}{5}\right) \approx 36.86^\circ \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distance Formula
In the world of geometry, understanding how far two points are from each other is key. The distance formula helps us do this easily. It is derived from the Pythagorean Theorem. For two points
  • \((x_1, y_1)\) and
  • \((x_2, y_2)\), we calculate the distance as: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]This formula helps determine the side lengths of triangles in coordinate geometry.
    For instance, to find side AB of a triangle with vertices A = (-1, 0) and B = (2, 1), we substitute into the formula:
    • \[AB = \sqrt{(2 - (-1))^2 + (1 - 0)^2} = \sqrt{3^2 + 1^2} = \sqrt{10}\]
      Similarly, finding the other sides with the distance formula allows us to fully describe the triangle in preparation for angle calculation. Precise distance understanding is crucial for further geometric analysis.

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Most popular questions from this chapter

In Exercises \(17-24\) , describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities. \begin{equation} \begin{array}{ll}{\text { a. } 0 \leq x \leq 1} & {\text { b. } 0 \leq x \leq 1, \quad 0 \leq y \leq 1} \\ {\text { c. } 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1}\end{array} \end{equation}

Write inequalities to describe the sets in Exercises \(45-50 .\) \begin{equation} \begin{array}{l}{\text { The closed region bounded by the spheres of radius } 1 \text { and radius } 2} \\ {\text { centered at the origin. (Closed means the spheres are to be included. }} \\ {\text { Had we wanted the spheres left out, we would have asked for }}\\\\{\text { the open region bounded by the spheres. This is analogous to the }} \\ {\text { way we use closed and open to describe intervals: closed means }} \\ {\text { endpoints included, open means endpoints left out. Closed sets }} \\ {\text { include boundaries; open sets leave them out. }}\end{array} \end{equation}

In Exercises \(1-16,\) give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$y=x^{2}, \quad z=0$$

Find the center \(C\) and the radius \(a\) for the spheres in Exercises \(51-60\) $$x^{2}+y^{2}+z^{2}+4 x-4 z=0$$

In Exercises \(17-24\) , describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities. \begin{equation} \text {a. }x \geq 0, \quad y \geq 0, \quad z=0 \quad \text { b. } x \geq 0, \quad y \leq 0, \quad z=0 \end{equation}
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