9. The figure shows a square with triangle ABC inside it. Ora 8 Vertices A and C are located at the midpoints of two sides of the square. What is the area of triangle ABC? A. 16 square units B. 24 square units C. 32 square units D. 40 square units
9. The figure shows a square with triangle ABC inside it. Ora 8 Vertices A and C are located at the midpoints of two sides of the square. What is the area of triangle ABC? A. 16 square units B. 24 square units C. 32 square units D. 40 square units
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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Question
Vertices A and C are located at the midpoints of two sides of the square. The side length of the square is given as 8 units. What is the area of triangle ABC?
**Multiple Choice Options:**
A. 16 square units
B. 24 square units
C. 32 square units
D. 40 square units
---
**Explanation of the Diagram:**
- The diagram consists of a square with side length labeled as 8 units.
- Inside the square, there is a triangle labeled ABC.
- Vertex A is located at the midpoint of the top side of the square.
- Vertex C is located at the midpoint of the right side of the square.
- Vertex B is located at the bottom-left corner of the square.
To find the area of triangle ABC, one needs to use the properties of the midpoints and the formula for the area of a triangle.
**Steps to Solve:**
1. **Identify the Coordinates:**
- Bottom-left corner, B: (0, 0)
- Midpoint of the top side, A: (4, 8)
- Midpoint of the right side, C: (8, 4)
2. **Use the Triangle Area Formula:**
The formula for the area of a triangle given its vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is:
\[
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
\]
3. **Substitute the Coordinates:**
\[
\text{Area} = \frac{1}{2} \left| 0(8 - 4) + 4(4 - 0) + 8(0 - 8) \right|
\]
\[
= \frac{1}{2} \left| 0 + 16 - 64 \right|
\]
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Transcribed Image Text:### Educational Website Content
---
#### Geometry Problem: Triangle within a Square
**Problem Statement:**
9. The figure shows a square with triangle ABC inside it.
Vertices A and C are located at the midpoints of two sides of the square. The side length of the square is given as 8 units. What is the area of triangle ABC?
**Multiple Choice Options:**
A. 16 square units
B. 24 square units
C. 32 square units
D. 40 square units
---
**Explanation of the Diagram:**
- The diagram consists of a square with side length labeled as 8 units.
- Inside the square, there is a triangle labeled ABC.
- Vertex A is located at the midpoint of the top side of the square.
- Vertex C is located at the midpoint of the right side of the square.
- Vertex B is located at the bottom-left corner of the square.
To find the area of triangle ABC, one needs to use the properties of the midpoints and the formula for the area of a triangle.
**Steps to Solve:**
1. **Identify the Coordinates:**
- Bottom-left corner, B: (0, 0)
- Midpoint of the top side, A: (4, 8)
- Midpoint of the right side, C: (8, 4)
2. **Use the Triangle Area Formula:**
The formula for the area of a triangle given its vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is:
\[
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
\]
3. **Substitute the Coordinates:**
\[
\text{Area} = \frac{1}{2} \left| 0(8 - 4) + 4(4 - 0) + 8(0 - 8) \right|
\]
\[
= \frac{1}{2} \left| 0 + 16 - 64 \right|
\]
\
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