4. For the three-part question that follows, provide your answer to each part in the given workspace. Identify each part with a coordinating response. Be sure to clearly label each part of your response as Part A, Part B, and Part C. Part A: State the Perpendicular Bisector Theorem. Part B: State the Converse to the Perpendicular Bisector Theorem. Part C: If AB = 7x, 7x, CB What is the length of CB? Show all Work B = 5x + 8, and BD 1 AC, CB =
4. For the three-part question that follows, provide your answer to each part in the given workspace. Identify each part with a coordinating response. Be sure to clearly label each part of your response as Part A, Part B, and Part C. Part A: State the Perpendicular Bisector Theorem. Part B: State the Converse to the Perpendicular Bisector Theorem. Part C: If AB = 7x, 7x, CB What is the length of CB? Show all Work B = 5x + 8, and BD 1 AC, CB =
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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![### Geometry Lesson: Understanding Perpendicular Bisectors
---
**Exercise 4: Perpendicular Bisectors and Their Theorems**
For the three-part question that follows, provide your answer to each part in the given workspace. Identify each part with a coordinating response. Be sure to clearly label each part of your response as Part A, Part B, and Part C.
#### Part A: State the Perpendicular Bisector Theorem.
The Perpendicular Bisector Theorem states that any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment.
#### Part B: State the Converse to the Perpendicular Bisector Theorem.
The Converse of the Perpendicular Bisector Theorem states that if a point is equidistant from the endpoints of a segment, then it lies on the perpendicular bisector of that segment.
#### Part C: Algebraic Application
Given:
\[ AB = 7x, \ CB = 5x + 8, \text{ and } \overline{BD} \perp \overline{AC} \]
What is the length of \( \overline{CB} \)?
**Show all work.**
**Solution:**
1. According to the problem, \( \overline{BD} \) bisects \( \overline{AC} \). Hence, the lengths of \( AB \) and \( CB \) should be equal.
2. Set up the equation based on the given lengths:
\[ 7x = 5x + 8 \]
3. Solve for \( x \):
\[ 7x - 5x = 8 \]
\[ 2x = 8 \]
\[ x = 4 \]
4. Substitute \( x = 4 \) back into the expression for \( CB \):
\[ CB = 5x + 8 \]
\[ CB = 5(4) + 8 \]
\[ CB = 20 + 8 \]
\[ CB = 28 \]
Therefore, the length of \( \overline{CB} \) is \( 28 \).
#### Diagram Explanation
The image shows an isosceles triangle \( \triangle ABC \) with point \( D \) on line \( AC \), where \( \overline{BD} \) is the perpendicular bisector of \( \overline{AC} \). The bisector intersects \( \overline{AC} \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9c2f64fc-e07a-43b3-a4b0-6f03d401f48c%2F67b82834-0a98-4d9b-bab6-037d4dceaea7%2Fyi23hm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Geometry Lesson: Understanding Perpendicular Bisectors
---
**Exercise 4: Perpendicular Bisectors and Their Theorems**
For the three-part question that follows, provide your answer to each part in the given workspace. Identify each part with a coordinating response. Be sure to clearly label each part of your response as Part A, Part B, and Part C.
#### Part A: State the Perpendicular Bisector Theorem.
The Perpendicular Bisector Theorem states that any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment.
#### Part B: State the Converse to the Perpendicular Bisector Theorem.
The Converse of the Perpendicular Bisector Theorem states that if a point is equidistant from the endpoints of a segment, then it lies on the perpendicular bisector of that segment.
#### Part C: Algebraic Application
Given:
\[ AB = 7x, \ CB = 5x + 8, \text{ and } \overline{BD} \perp \overline{AC} \]
What is the length of \( \overline{CB} \)?
**Show all work.**
**Solution:**
1. According to the problem, \( \overline{BD} \) bisects \( \overline{AC} \). Hence, the lengths of \( AB \) and \( CB \) should be equal.
2. Set up the equation based on the given lengths:
\[ 7x = 5x + 8 \]
3. Solve for \( x \):
\[ 7x - 5x = 8 \]
\[ 2x = 8 \]
\[ x = 4 \]
4. Substitute \( x = 4 \) back into the expression for \( CB \):
\[ CB = 5x + 8 \]
\[ CB = 5(4) + 8 \]
\[ CB = 20 + 8 \]
\[ CB = 28 \]
Therefore, the length of \( \overline{CB} \) is \( 28 \).
#### Diagram Explanation
The image shows an isosceles triangle \( \triangle ABC \) with point \( D \) on line \( AC \), where \( \overline{BD} \) is the perpendicular bisector of \( \overline{AC} \). The bisector intersects \( \overline{AC} \)
Transcribed Image Text:# Equilateral Triangle Analysis
### **Show all Work**
#### Diagram Description:
The image displays an equilateral triangle inscribed with key points and symmetry lines.
- **Vertices**: The triangle is labeled with three vertices: A, B, and C.
- **Interior Line**: A line is drawn from vertex B perpendicular to the base AC, intersecting at point D, creating two right-angle triangles ABD and BDC.
- **Equal Sides**: The sides AB and BC are marked with small lines, indicating that they are of equal length.
#### Analysis:
In an equilateral triangle:
1. **All sides are of equal length**.
2. **Each angle measures 60 degrees**.
3. **The perpendicular from the vertex opposite the base bisects the base and the vertex angle**, creating two 30-60-90 triangles.
**Key Observations in this Diagram:**
- **Perpendicular Bisector**: Line BD is a perpendicular bisector, meaning it equally divides the base AC into two equal segments, AD and DC.
- **Symmetry**: This creates two congruent right triangles: ΔABD and ΔBDC.
- **Properties of Right Triangles ABD and BDC**:
- \(\angle ADB = \angle CDB = 90^\circ\)
- \(\angle DAB = \angle DBC = 60^\circ\)
- \(\angle ABD = \angle CBD = 30^\circ\)
#### Mathematical Implications:
Using properties of 30-60-90 triangles:
1. If the length of BD (height from B to AC) is \(h\), then:
- \(AD = DC = \frac{AC}{2}\)
- \(AB = BC = \sqrt{ (\frac{AC}{2})^2 + h^2 }\)
### Further Exploration:
Students can use this diagram to:
- Calculate the lengths of the sides using trigonometric relationships.
- Prove that the triangle is equilateral by confirming that all sides and angles are equal.
- Explore the area of the triangle by splitting it into two right triangles.
### Practice Problem:
If side \(AC = 8\) and point D is the midpoint, calculate lengths AB and BD.
Ensure students to show all work for full credit.
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