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Chapter 14: Problem 1279

\(\mathrm{A}(4,0), \mathrm{B}(0,3), \mathrm{C}(6,1)\) be vertices of triangle \(\mathrm{ABC}\). Slope of bisector of angle \(\mathrm{C}\) will be (a) \(3 \sqrt{2}-7\) (b) \(5 \sqrt{2}-7\) (c) \(6 \sqrt{2}-7\) (d) none

Short Answer

Expert verified
The slope of the angle bisector at vertex C is \(\frac{\sqrt{200} - 200}{(8\sqrt{5}+\sqrt{40})}\), which does not match any of the given options. Therefore, the correct answer is (d) none.

Step by step solution

01

1. Calculate the lengths of sides AC and BC

First, we need to find the lengths of sides AC and BC to apply the angle bisector formula. We will use the distance formula to find the lengths: \( AC = \sqrt{(6-4)^2 + (1-0)^2} = \sqrt{2^2 + 1^2} = \sqrt{5} \) \( BC = \sqrt{(0-6)^2 + (3-1)^2} = \sqrt{(-6)^2 + 2^2} = \sqrt{40} \)
02

2. Apply the angle bisector formula

Now that we have the lengths of AC and BC, we can find the angle bisector's slope using the following formula: \( tan(\text{bisector}\ angle\ BAC) = \frac{m_Am_B - 1}{m_A + m_B} \) where \(m_A\) and \(m_B\) are given by: \( m_A = \frac{\sqrt{5}}{5} \) \( m_B = \frac{\sqrt{40}}{40} \)
03

3. Calculate the slope of angle bisector

Now we can plug \(m_A\) and \(m_B\) into the bisector angle formula: \( tan(\text{bisector}\ angle\ BAC) = \frac{\frac{\sqrt{5}}{5}*\frac{\sqrt{40}}{40} - 1}{\frac{\sqrt{5}}{5} + \frac{\sqrt{40}}{40}} \) Simplify the expression: \( tan(\text{bisector}\ angle\ BAC) = \frac{\frac{\sqrt{200}}{200} - 1}{\frac{8\sqrt{5} + \sqrt{40}}{40}} \) Multiply both the numerator and denominator by 200 to further simplify: \( tan(\text{bisector}\ angle\ BAC) = \frac{\sqrt{200} - 200}{(8\sqrt{5}+\sqrt{40})} \)
04

4. Compare the result with the given options

Now compare the simplified expression of the angle bisector's slope with the given options: (a) \(3 \sqrt{2}-7\) (b) \(5 \sqrt{2}-7\) (c) \(6 \sqrt{2}-7\) (d) none None of the given options match our answer, so the correct option is: (d) none

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