4. Triangle ABC is a right triangle with right angle ZBCA. Assume AD = DB,DE I AB, AB = 20m, and AC =12m. Determine the area of quadrilateral ADEC. E A B
4. Triangle ABC is a right triangle with right angle ZBCA. Assume AD = DB,DE I AB, AB = 20m, and AC =12m. Determine the area of quadrilateral ADEC. E A B
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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![### Problem Statement
4. Triangle \( \triangle ABC \) is a right triangle with a right angle at \( \angle BCA \). Assume \( AD \cong DB \), \( DE \perp AB \), \( AB = 20m \), and \( AC = 12m \). Determine the area of quadrilateral \( ADEC \).
### Diagram Explanation
The diagram illustrates a right triangle \( \triangle ABC \) with the following features:
- Vertex \( C \) is the right angle.
- Segment \( AD \) is congruent to segment \( DB \), indicating that \( D \) is the midpoint of \( AB \).
- Line \( DE \) is perpendicular to \( AB \) at point \( D \), forming a small right triangle \( \triangle CDE \) within the larger triangle.
- Points \( A \), \( D \), \( E \), and \( C \) form the quadrilateral \( ADEC \), whose area needs to be determined.
### Steps to Solution
1. **Identify Known Lengths:**
- \( AB = 20m \)
- \( AC = 12m \)
- Since \( D \) is the midpoint of \( AB \), \( AD = DB = 10m \).
2. **Calculate Area of Triangle \( \triangle ABC \):**
Since \( \triangle ABC \) is a right triangle,
\[
\text{Area of } \triangle ABC = \frac{1}{2} \times AB \times AC = \frac{1}{2} \times 20m \times 12m = 120m^2.
\]
3. **Calculate the Area of Triangle \( \triangle CDE \):**
Since \( DE \perp AB \),
\[
\text{Height } DE = 12m \text{ (same as side AC)}
\]
\[
\text{Area of } \triangle CDE = \frac{1}{2} \times DE \times DB = \frac{1}{2} \times 12m \times 10m = 60m^2.
\]
4. **Determine Area of Quadrilateral \( ADEC \):**
Since quadrilateral \( ADEC \) consists of the area of triangle \( \triangle ABC \) minus the area of triangle \( \triangle CDE](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff93238f1-b105-487d-8b56-1173b2b7a646%2Fc07d590b-0046-4824-b954-1608a57201d8%2F10m90a_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
4. Triangle \( \triangle ABC \) is a right triangle with a right angle at \( \angle BCA \). Assume \( AD \cong DB \), \( DE \perp AB \), \( AB = 20m \), and \( AC = 12m \). Determine the area of quadrilateral \( ADEC \).
### Diagram Explanation
The diagram illustrates a right triangle \( \triangle ABC \) with the following features:
- Vertex \( C \) is the right angle.
- Segment \( AD \) is congruent to segment \( DB \), indicating that \( D \) is the midpoint of \( AB \).
- Line \( DE \) is perpendicular to \( AB \) at point \( D \), forming a small right triangle \( \triangle CDE \) within the larger triangle.
- Points \( A \), \( D \), \( E \), and \( C \) form the quadrilateral \( ADEC \), whose area needs to be determined.
### Steps to Solution
1. **Identify Known Lengths:**
- \( AB = 20m \)
- \( AC = 12m \)
- Since \( D \) is the midpoint of \( AB \), \( AD = DB = 10m \).
2. **Calculate Area of Triangle \( \triangle ABC \):**
Since \( \triangle ABC \) is a right triangle,
\[
\text{Area of } \triangle ABC = \frac{1}{2} \times AB \times AC = \frac{1}{2} \times 20m \times 12m = 120m^2.
\]
3. **Calculate the Area of Triangle \( \triangle CDE \):**
Since \( DE \perp AB \),
\[
\text{Height } DE = 12m \text{ (same as side AC)}
\]
\[
\text{Area of } \triangle CDE = \frac{1}{2} \times DE \times DB = \frac{1}{2} \times 12m \times 10m = 60m^2.
\]
4. **Determine Area of Quadrilateral \( ADEC \):**
Since quadrilateral \( ADEC \) consists of the area of triangle \( \triangle ABC \) minus the area of triangle \( \triangle CDE
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