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A 8. C 7 6 4 3 2 -7 -6 -5 -2 -1 0 2 3 7 8 10 11 -1 A -2 -3 3 vertices of a parallelogram ABCD are shown in the applet. What are the coordinates of point D?

A 8. C 7 6 4 3 2 -7 -6 -5 -2 -1 0 2 3 7 8 10 11 -1 A -2 -3 3 vertices of a parallelogram ABCD are shown in the applet. What are the coordinates of point D?

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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### Identifying the Coordinates of Point D in a Parallelogram **Introduction:** In this task, we are given three vertices of a parallelogram \(ABCD\) plotted on a coordinate plane. Our goal is to identify the coordinates of the fourth vertex, point \(D\). **Graph Analysis:** - **Point \(A\)** is located at \( (4, -1) \). - **Point \(B\)** is situated at \( (8, 3) \). - **Point \(C\)** is positioned at \( (2, 8) \). The graph represents a coordinate plane with the x-axis ranging approximately from \(-7\) to \(11\) and the y-axis ranging approximately from \(-3\) to \(8\). The points are plotted and labeled accordingly: \(A\), \(B\), and \(C\). **Problem Statement:** Using the given points of the parallelogram, determine the coordinates of the fourth vertex, point \(D\). **Solution Approach:** 1. **Identify vectors:** - Vector \(\overrightarrow{AB}\) moves from \( (4, -1) \) to \( (8, 3) \). Therefore, \(\overrightarrow{AB} = (8 - 4, 3 + 1) = (4, 4)\). - Vector \(\overrightarrow{AD}\) must be equal to \(\overrightarrow{BC}\). 2. **Compute Vector \(\overrightarrow{BC}\):** - \(\overrightarrow{BC}\) moves from \( (8, 3) \) to \( (2, 8) \). Hence, \(\overrightarrow{BC} = (2 - 8, 8 - 3) = (-6, 5)\). 3. **Determine coordinates of \(D\):** - Since vector \(\overrightarrow{AD} = \overrightarrow{BC}\), then adding \(\overrightarrow{BC}\) to point \(A\): - Coordinates of \(D\) = \(A + (-6, 5)\) = \( (4 - 6, -1 + 5) \) = \((-2, 4)\). **Conclusion:** The coordinates of point \(D\) in the parallelogram \(ABCD\) are \((-2
Transcribed Image Text:### Identifying the Coordinates of Point D in a Parallelogram **Introduction:** In this task, we are given three vertices of a parallelogram \(ABCD\) plotted on a coordinate plane. Our goal is to identify the coordinates of the fourth vertex, point \(D\). **Graph Analysis:** - **Point \(A\)** is located at \( (4, -1) \). - **Point \(B\)** is situated at \( (8, 3) \). - **Point \(C\)** is positioned at \( (2, 8) \). The graph represents a coordinate plane with the x-axis ranging approximately from \(-7\) to \(11\) and the y-axis ranging approximately from \(-3\) to \(8\). The points are plotted and labeled accordingly: \(A\), \(B\), and \(C\). **Problem Statement:** Using the given points of the parallelogram, determine the coordinates of the fourth vertex, point \(D\). **Solution Approach:** 1. **Identify vectors:** - Vector \(\overrightarrow{AB}\) moves from \( (4, -1) \) to \( (8, 3) \). Therefore, \(\overrightarrow{AB} = (8 - 4, 3 + 1) = (4, 4)\). - Vector \(\overrightarrow{AD}\) must be equal to \(\overrightarrow{BC}\). 2. **Compute Vector \(\overrightarrow{BC}\):** - \(\overrightarrow{BC}\) moves from \( (8, 3) \) to \( (2, 8) \). Hence, \(\overrightarrow{BC} = (2 - 8, 8 - 3) = (-6, 5)\). 3. **Determine coordinates of \(D\):** - Since vector \(\overrightarrow{AD} = \overrightarrow{BC}\), then adding \(\overrightarrow{BC}\) to point \(A\): - Coordinates of \(D\) = \(A + (-6, 5)\) = \( (4 - 6, -1 + 5) \) = \((-2, 4)\). **Conclusion:** The coordinates of point \(D\) in the parallelogram \(ABCD\) are \((-2
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