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The image shows quadrilateral ABCD. Find the Area of ABCD. 7 1 2 3 4 5 6 7 X 6. 5. 4) 3. 2.

The image shows quadrilateral ABCD. Find the Area of ABCD. 7 1 2 3 4 5 6 7 X 6. 5. 4) 3. 2.

Elementary Geometry For College Students, 7e
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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The image shows quadrilateral ABCD. Find the area and perimeter of ABCD.

**Quadrilateral ABCD Area Calculation** In this exercise, we are tasked with finding the area of quadrilateral ABCD, which is plotted on a coordinate grid. ### Coordinate Points: The vertices of quadrilateral ABCD are: - \( A (2, 4) \) - \( B (3, 1) \) - \( C (6, 2) \) - \( D (5, 7) \) ### Graph Description: The graph consists of a coordinate plane with points labeled \( A \), \( B \), \( C \), and \( D \) connected to form the quadrilateral. The x-axis ranges from 0 to 7 and the y-axis ranges from 0 to 7. Each grid square represents one unit. ### Finding the Area: To find the area of quadrilateral ABCD, we can apply the Shoelace Theorem (also known as Gauss's area formula for polygons). The formula for the area based on vertex coordinates \((x_1, y_1), (x_2, y_2), ..., (x_n, y_n)\) is given by: \[ Area = \frac{1}{2} \left| \sum_{i=1}^{n-1} (x_i y_{i+1} - y_i x_{i+1}) + (x_n y_1 - y_n x_1) \right| \] Substituting the coordinates of the vertices: \[ \begin{aligned} &Area = \frac{1}{2} \left| (2 \cdot 1 + 3 \cdot 2 + 6 \cdot 7 + 5 \cdot 4) - (4 \cdot 3 + 1 \cdot 6 + 2 \cdot 5 + 7 \cdot 2) \right| \\ &= \frac{1}{2} \left| (2 + 6 + 42 + 20) - (12 + 6 + 10 + 14) \right| \\ &= \frac{1}{2} \left| 70 - 42 \right| \\ &= \frac{1}{2} \cdot 28 \\ &= 14 \end{aligned} \] Therefore, the area of quadrilateral ABCD is **14 square
Transcribed Image Text:**Quadrilateral ABCD Area Calculation** In this exercise, we are tasked with finding the area of quadrilateral ABCD, which is plotted on a coordinate grid. ### Coordinate Points: The vertices of quadrilateral ABCD are: - \( A (2, 4) \) - \( B (3, 1) \) - \( C (6, 2) \) - \( D (5, 7) \) ### Graph Description: The graph consists of a coordinate plane with points labeled \( A \), \( B \), \( C \), and \( D \) connected to form the quadrilateral. The x-axis ranges from 0 to 7 and the y-axis ranges from 0 to 7. Each grid square represents one unit. ### Finding the Area: To find the area of quadrilateral ABCD, we can apply the Shoelace Theorem (also known as Gauss's area formula for polygons). The formula for the area based on vertex coordinates \((x_1, y_1), (x_2, y_2), ..., (x_n, y_n)\) is given by: \[ Area = \frac{1}{2} \left| \sum_{i=1}^{n-1} (x_i y_{i+1} - y_i x_{i+1}) + (x_n y_1 - y_n x_1) \right| \] Substituting the coordinates of the vertices: \[ \begin{aligned} &Area = \frac{1}{2} \left| (2 \cdot 1 + 3 \cdot 2 + 6 \cdot 7 + 5 \cdot 4) - (4 \cdot 3 + 1 \cdot 6 + 2 \cdot 5 + 7 \cdot 2) \right| \\ &= \frac{1}{2} \left| (2 + 6 + 42 + 20) - (12 + 6 + 10 + 14) \right| \\ &= \frac{1}{2} \left| 70 - 42 \right| \\ &= \frac{1}{2} \cdot 28 \\ &= 14 \end{aligned} \] Therefore, the area of quadrilateral ABCD is **14 square
**Understanding the Perimeter of a Quadrilateral** The image shows quadrilateral ABCD plotted on a coordinate grid. Our goal is to find the perimeter of quadrilateral ABCD. ### Points and Coordinates The vertices of the quadrilateral are labeled as follows: - Point A at (2, 4) - Point B at (3, 2) - Point C at (7, 3) - Point D at (6, 7) ### Explanation of the Graph The graph displays a Cartesian coordinate system with the x-axis and y-axis. Each unit represents one coordinate point. ### Finding the Perimeter To determine the perimeter, we need to calculate the length of each side of quadrilateral ABCD. We use the distance formula to find the distance between any two points \((x_1, y_1)\) and \((x_2, y_2)\): \[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] #### 1. Length of AB Coordinates: A(2, 4) and B(3, 2) \[ AB = \sqrt{(3 - 2)^2 + (2 - 4)^2} = \sqrt{1 + 4} = \sqrt{5} \approx 2.24 \] #### 2. Length of BC Coordinates: B(3, 2) and C(7, 3) \[ BC = \sqrt{(7 - 3)^2 + (3 - 2)^2} = \sqrt{16 + 1} = \sqrt{17} \approx 4.12 \] #### 3. Length of CD Coordinates: C(7, 3) and D(6, 7) \[ CD = \sqrt{(6 - 7)^2 + (7 - 3)^2} = \sqrt{1 + 16} = \sqrt{17} \approx 4.12 \] #### 4. Length of DA Coordinates: D(6, 7) and A(2, 4) \[ DA = \sqrt{(2 - 6)^2 + (4 - 7)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \] ### Calculating the Perimeter
Transcribed Image Text:**Understanding the Perimeter of a Quadrilateral** The image shows quadrilateral ABCD plotted on a coordinate grid. Our goal is to find the perimeter of quadrilateral ABCD. ### Points and Coordinates The vertices of the quadrilateral are labeled as follows: - Point A at (2, 4) - Point B at (3, 2) - Point C at (7, 3) - Point D at (6, 7) ### Explanation of the Graph The graph displays a Cartesian coordinate system with the x-axis and y-axis. Each unit represents one coordinate point. ### Finding the Perimeter To determine the perimeter, we need to calculate the length of each side of quadrilateral ABCD. We use the distance formula to find the distance between any two points \((x_1, y_1)\) and \((x_2, y_2)\): \[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] #### 1. Length of AB Coordinates: A(2, 4) and B(3, 2) \[ AB = \sqrt{(3 - 2)^2 + (2 - 4)^2} = \sqrt{1 + 4} = \sqrt{5} \approx 2.24 \] #### 2. Length of BC Coordinates: B(3, 2) and C(7, 3) \[ BC = \sqrt{(7 - 3)^2 + (3 - 2)^2} = \sqrt{16 + 1} = \sqrt{17} \approx 4.12 \] #### 3. Length of CD Coordinates: C(7, 3) and D(6, 7) \[ CD = \sqrt{(6 - 7)^2 + (7 - 3)^2} = \sqrt{1 + 16} = \sqrt{17} \approx 4.12 \] #### 4. Length of DA Coordinates: D(6, 7) and A(2, 4) \[ DA = \sqrt{(2 - 6)^2 + (4 - 7)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \] ### Calculating the Perimeter
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