Problem 83
Question
Let \(P=(-1,0), Q=(0,0)\) and \(R=(3,3 \sqrt{3})\) be three points. The equation of the bisector of the angle \(P Q R\) (A) \(\sqrt{3} x+y=0\) (B) \(x+\frac{\sqrt{3}}{2} y=0\) (C) \(\frac{\sqrt{3}}{2} x+y=0\) (D) \(x+\sqrt{3} y=0\)
Step-by-Step Solution
Verified Answer
The equation of the bisector is (D) \(x + \sqrt{3}y = 0\).
1Step 1: Determine Slopes of PQ and QR
To find the bisector of angle \(PQR\), first determine the slopes of segments \(PQ\) and \(QR\). The slope of \(PQ\), with points \(P(-1, 0)\) and \(Q(0, 0)\), is \(m_{PQ} = \frac{0-0}{0+1} = 0\). The slope of \(QR\), with points \(Q(0, 0)\) and \(R(3, 3\sqrt{3})\), is \(m_{QR} = \frac{3\sqrt{3}-0}{3-0} = \sqrt{3}\).
2Step 2: Calculate Angle Bisector Slope
The angle bisector of two lines with slopes \(m_1\) and \(m_2\) has a slope given by \(m = \frac{m_1 + m_2 - |m_1 m_2 - 1|}{1 + m_1 m_2 + |m_1 - m_2|}\) or \(m = \frac{m_1 + m_2 + |m_1 m_2 - 1|}{1 + m_1 m_2 - |m_1 - m_2|}\). Substitute \(m_1 = 0\) and \(m_2 = \sqrt{3}\).
3Step 3: Solve for Slope of Angle Bisector
Substitute values into the formula: \(m = \frac{0 + \sqrt{3} - |0 \times \sqrt{3} - 1|}{1 + 0 \times \sqrt{3} + |0 - \sqrt{3}|} = \frac{\sqrt{3} - 1}{1 + \sqrt{3}}\). Simplifying gives the slope of the bisector of angle \(PQR\) as \(-\frac{1}{\sqrt{3}}\).
4Step 4: Write Equation of Bisector Line
The general equation of a line with slope \(m\) passing through a point (e.g., \(Q(0,0)\)) is \(y = mx + c\). Since \(c = 0\) when \(y\) and \(x\) are zero, the equation is \(y = -\frac{1}{\sqrt{3}}x\). Multiply through by \(\sqrt{3}\) to get \(\sqrt{3}y = -x\) or \(x + \sqrt{3}y = 0\).
5Step 5: Identify the Correct Option
From the given options, we identify the equation of the angle bisector as \(x + \sqrt{3}y = 0\). Compare this with the available choices.
Key Concepts
Slope of a LineEquation of a LineAngle Geometry
Slope of a Line
The slope of a line is a measure of how steep the line is, typically represented by the letter \(m\). It tells us how much the line rises or falls as it moves from left to right.
If we have a line connecting two points, \((x_1, y_1)\) and \((x_2, y_2)\), the slope is calculated using the formula:
In our problem, examining segments \(PQ\) (from \(P(-1, 0)\) to \(Q(0, 0)\)) and \(QR\) (from \(Q(0, 0)\) to \(R(3, 3\sqrt{3})\)), we can find their slopes. Since both \(P\) and \(Q\) lie on the x-axis, the slope of \(PQ\) is 0, indicating a horizontal line.
Next, for \(QR\), the slope is calculated to be \(\sqrt{3}\). This positive value shows the line rises upwards to the right at an angle.
If we have a line connecting two points, \((x_1, y_1)\) and \((x_2, y_2)\), the slope is calculated using the formula:
- \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
In our problem, examining segments \(PQ\) (from \(P(-1, 0)\) to \(Q(0, 0)\)) and \(QR\) (from \(Q(0, 0)\) to \(R(3, 3\sqrt{3})\)), we can find their slopes. Since both \(P\) and \(Q\) lie on the x-axis, the slope of \(PQ\) is 0, indicating a horizontal line.
Next, for \(QR\), the slope is calculated to be \(\sqrt{3}\). This positive value shows the line rises upwards to the right at an angle.
Equation of a Line
The equation of a line in slope-intercept form is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept, which is where the line crosses the y-axis.
In the context of our exercise, we want the line that is the bisector of angle \(PQR\).
This means it perfectly divides angle \(PQR\) into two equal halves. Once we calculated the slope of the angle bisector as \(-\frac{1}{\sqrt{3}}\), we need an equation that passes through point \(Q(0, 0)\).
The formula becomes \(y = -\frac{1}{\sqrt{3}}x\). Since the line passes through the origin, the y-intercept \(b\) is zero.
To write it in a neater form, multiply both sides by \(\sqrt{3}\) to remove the fraction, resulting in \(x + \sqrt{3}y = 0\). This is the equation of the angle bisector line.
In the context of our exercise, we want the line that is the bisector of angle \(PQR\).
This means it perfectly divides angle \(PQR\) into two equal halves. Once we calculated the slope of the angle bisector as \(-\frac{1}{\sqrt{3}}\), we need an equation that passes through point \(Q(0, 0)\).
The formula becomes \(y = -\frac{1}{\sqrt{3}}x\). Since the line passes through the origin, the y-intercept \(b\) is zero.
To write it in a neater form, multiply both sides by \(\sqrt{3}\) to remove the fraction, resulting in \(x + \sqrt{3}y = 0\). This is the equation of the angle bisector line.
Angle Geometry
Angle geometry involves understanding the measure and relationships of angles, particularly in geometric shapes and figures.
In our scenario, angle geometry is crucial for discovering how the angle bisector functions. First, recall the angle bisector theorem: It states that the angle bisector divides the opposite side into two segments that are proportional to the other two sides.
For lines with slopes \(m_1 = 0\) and \(m_2 = \sqrt{3}\), the formula can find the slope of the angle bisector.
The bisector cuts through the angle \(PQR\) equally, providing us with the needed line equation.
In our scenario, angle geometry is crucial for discovering how the angle bisector functions. First, recall the angle bisector theorem: It states that the angle bisector divides the opposite side into two segments that are proportional to the other two sides.
For lines with slopes \(m_1 = 0\) and \(m_2 = \sqrt{3}\), the formula can find the slope of the angle bisector.
- To calculate, use the given bisector formula with slopes: \(m = \frac{m_1 + m_2 - |m_1m_2 - 1|}{1 + m_1m_2 + |m_1 - m_2|}\).
The bisector cuts through the angle \(PQR\) equally, providing us with the needed line equation.
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