ASTU has vertices S(-6-3). T(5.4), and U(7.-4). Write the equation of the perpendicular bisector of side TU.
ASTU has vertices S(-6-3). T(5.4), and U(7.-4). Write the equation of the perpendicular bisector of side TU.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![**Problem 6:**
Triangle \( \triangle STU \) has vertices \( S(-6, -3) \), \( T(5, 4) \), and \( U(7, -4) \). Write the equation of the perpendicular bisector of side \( TU \).
**Explanation:**
To find the perpendicular bisector of side \( TU \), follow these steps:
1. **Find the Midpoint of \( TU \):**
\[
\text{Midpoint} = \left( \frac{5 + 7}{2}, \frac{4 + (-4)}{2} \right) = (6, 0)
\]
2. **Calculate the Slope of \( TU \):**
\[
\text{Slope of } TU = \frac{-4 - 4}{7 - 5} = \frac{-8}{2} = -4
\]
3. **Determine the Perpendicular Slope:**
The negative reciprocal of \(-4\) is \(\frac{1}{4}\).
4. **Write the Equation of the Perpendicular Bisector:**
Using the point-slope form \( y - y_1 = m(x - x_1) \), with midpoint \((6, 0)\) and slope \(\frac{1}{4}\):
\[
y - 0 = \frac{1}{4}(x - 6)
\]
Simplifying, the equation is:
\[
y = \frac{1}{4}x - \frac{3}{2}
\]
**Graph Description:**
Below the text, there is a grid representing a standard Cartesian coordinate system with axes. The grid can be used to plot the triangle's vertices and visualize the perpendicular bisector graphically.
**Visual Aid:**
The image also includes a small graphic of interlocking puzzle pieces, which may symbolize problem-solving or piecing together concepts in geometry.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd6ac5d52-bb7d-4d94-837c-772677a7135e%2F1692708e-1a99-4778-8435-051b87a928ca%2Ftvmrtm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 6:**
Triangle \( \triangle STU \) has vertices \( S(-6, -3) \), \( T(5, 4) \), and \( U(7, -4) \). Write the equation of the perpendicular bisector of side \( TU \).
**Explanation:**
To find the perpendicular bisector of side \( TU \), follow these steps:
1. **Find the Midpoint of \( TU \):**
\[
\text{Midpoint} = \left( \frac{5 + 7}{2}, \frac{4 + (-4)}{2} \right) = (6, 0)
\]
2. **Calculate the Slope of \( TU \):**
\[
\text{Slope of } TU = \frac{-4 - 4}{7 - 5} = \frac{-8}{2} = -4
\]
3. **Determine the Perpendicular Slope:**
The negative reciprocal of \(-4\) is \(\frac{1}{4}\).
4. **Write the Equation of the Perpendicular Bisector:**
Using the point-slope form \( y - y_1 = m(x - x_1) \), with midpoint \((6, 0)\) and slope \(\frac{1}{4}\):
\[
y - 0 = \frac{1}{4}(x - 6)
\]
Simplifying, the equation is:
\[
y = \frac{1}{4}x - \frac{3}{2}
\]
**Graph Description:**
Below the text, there is a grid representing a standard Cartesian coordinate system with axes. The grid can be used to plot the triangle's vertices and visualize the perpendicular bisector graphically.
**Visual Aid:**
The image also includes a small graphic of interlocking puzzle pieces, which may symbolize problem-solving or piecing together concepts in geometry.
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