Question
Your teacher will grade your responses for questions 7–9 to ensure that you receive proper credit for your answers.
Given triangle GHI with [point G at ( 4, –3), point H at (–4, 2), and I at (2, 4), find the perpendicular bisector of segment H I in standard form.
(2 points)
Given triangle GHI with [point G at ( 4, –3), point H at (–4, 2), and I at (2, 4), find the perpendicular bisector of segment H I in standard form.
(2 points)
Answers
GPT-4 Turbo
To find the perpendicular bisector of segment HI in a triangle GHI, we need to perform the following steps:
1. Find the midpoint of segment HI.
2. Determine the slope of segment HI.
3. Use the negative reciprocal of that slope to get the slope of the perpendicular bisector because perpendicular lines have slopes that are negative reciprocals of each other.
4. Use the midpoint and the slope of the perpendicular bisector to formulate the equation of the bisector line.
Let's follow these steps:
**Step 1:** Find the midpoint of segment HI.
The midpoint M of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\).
For HI with \(H(-4, 2)\) and \(I(2, 4)\), the midpoint \(M\) is:
\(M = \left(\frac{-4 + 2}{2}, \frac{2 + 4}{2}\right)\)
\(M = \left(\frac{-2}{2}, \frac{6}{2}\right)\)
\(M = (-1, 3)\)
**Step 2:** Determine the slope of segment HI.
The slope of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
For HI, we have:
\(m_{HI} = \frac{4 - 2}{2 - (-4)}\)
\(m_{HI} = \frac{2}{6}\)
\(m_{HI} = \frac{1}{3}\)
**Step 3:** Find the negative reciprocal of the slope of HI for the perpendicular bisector's slope.
\(m_{perpendicular} = -\frac{1}{m_{HI}}\)
\(m_{perpendicular} = -\frac{1}{\frac{1}{3}}\)
\(m_{perpendicular} = -3\)
**Step 4:** Write the equation of the perpendicular bisector using the point-slope form.
The point-slope form is given by \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a point on the line and \(m\) is the slope.
Using midpoint \(M(-1, 3)\) and slope \(m_{perpendicular} = -3\), we have:
\(y - 3 = -3(x + 1)\)
Now we can rewrite this into standard form, which is \(Ax + By = C\).
\(y + 3x + 3 = 3\)
Bringing all terms to one side, we get:
\(3x - y = 0\)
Therefore, the perpendicular bisector of segment HI in standard form is \(3x - y = 0\).
1. Find the midpoint of segment HI.
2. Determine the slope of segment HI.
3. Use the negative reciprocal of that slope to get the slope of the perpendicular bisector because perpendicular lines have slopes that are negative reciprocals of each other.
4. Use the midpoint and the slope of the perpendicular bisector to formulate the equation of the bisector line.
Let's follow these steps:
**Step 1:** Find the midpoint of segment HI.
The midpoint M of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\).
For HI with \(H(-4, 2)\) and \(I(2, 4)\), the midpoint \(M\) is:
\(M = \left(\frac{-4 + 2}{2}, \frac{2 + 4}{2}\right)\)
\(M = \left(\frac{-2}{2}, \frac{6}{2}\right)\)
\(M = (-1, 3)\)
**Step 2:** Determine the slope of segment HI.
The slope of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
For HI, we have:
\(m_{HI} = \frac{4 - 2}{2 - (-4)}\)
\(m_{HI} = \frac{2}{6}\)
\(m_{HI} = \frac{1}{3}\)
**Step 3:** Find the negative reciprocal of the slope of HI for the perpendicular bisector's slope.
\(m_{perpendicular} = -\frac{1}{m_{HI}}\)
\(m_{perpendicular} = -\frac{1}{\frac{1}{3}}\)
\(m_{perpendicular} = -3\)
**Step 4:** Write the equation of the perpendicular bisector using the point-slope form.
The point-slope form is given by \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a point on the line and \(m\) is the slope.
Using midpoint \(M(-1, 3)\) and slope \(m_{perpendicular} = -3\), we have:
\(y - 3 = -3(x + 1)\)
Now we can rewrite this into standard form, which is \(Ax + By = C\).
\(y + 3x + 3 = 3\)
Bringing all terms to one side, we get:
\(3x - y = 0\)
Therefore, the perpendicular bisector of segment HI in standard form is \(3x - y = 0\).