Pregunta
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Rectangle \( A B C D \) has an area of 18 square units with \( A(-5,5) \) and \( B(1,5) \) 1. What are the dimensions of the rectangle? 2. What are the remaining two points that form the rectangle? For full credit, you must: - Answer all parts of the question. - Show your work - Show any formulas that you used.

Ask by Bird George. in the United States
Jan 23,2025

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The rectangle has a length of 6 units and a width of 3 units. The remaining two points are \( C(1, 8) \) and \( D(-5, 8) \).

Solución

Let's solve the problem step by step, addressing each part as required. **Given:** - Rectangle \( ABCD \) has an area of 18 square units. - Points \( A(-5, 5) \) and \( B(1, 5) \). ### 1. **Finding the Dimensions of the Rectangle** **Step 1: Calculate the Length of Side \( AB \)** Since points \( A \) and \( B \) have the same \( y \)-coordinate, \( AB \) is a horizontal side. - **Coordinates:** - \( A(-5, 5) \) - \( B(1, 5) \) - **Length Formula (Distance between two points on a horizontal line):** \[ \text{Length} = |x_2 - x_1| \] - **Calculation:** \[ \text{Length } AB = |1 - (-5)| = |1 + 5| = 6 \text{ units} \] **Step 2: Determine the Width Using the Area** - **Area Formula for Rectangle:** \[ \text{Area} = \text{Length} \times \text{Width} \] - **Given Area:** \[ 18 = 6 \times \text{Width} \] - **Solve for Width:** \[ \text{Width} = \frac{18}{6} = 3 \text{ units} \] **Answer to Part 1:** - **Dimensions of the Rectangle:** Length = 6 units, Width = 3 units. ### 2. **Finding the Remaining Two Points \( C \) and \( D \)** **Understanding the Rectangle's Orientation:** - Points \( A \) and \( B \) are horizontally aligned. - The rectangle can extend either **upwards** or **downwards** from side \( AB \). We'll consider both possibilities. **Case 1: Rectangle Extends Upwards (Positive \( y \)-direction)** - **Point \( D \):** Located directly above \( A \). \[ D = (-5, 5 + 3) = (-5, 8) \] - **Point \( C \):** Located directly above \( B \). \[ C = (1, 5 + 3) = (1, 8) \] **Case 2: Rectangle Extends Downwards (Negative \( y \)-direction)** - **Point \( D \):** Located directly below \( A \). \[ D = (-5, 5 - 3) = (-5, 2) \] - **Point \( C \):** Located directly below \( B \). \[ C = (1, 5 - 3) = (1, 2) \] **Conclusion:** Without additional information on the rectangle's orientation, both cases are valid. However, typically, we assume the rectangle extends upwards unless specified otherwise. **Answer to Part 2:** - **Remaining Points:** - \( C(1, 8) \) - \( D(-5, 8) \) **Summary:** 1. **Dimensions:** Length = 6 units, Width = 3 units. 2. **Remaining Points:** \( C(1, 8) \) and \( D(-5, 8) \).

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Mind Expander

To solve for the dimensions and the remaining points of rectangle \( ABCD \), we will start by finding the length of sides and then determine the coordinates of the remaining vertices. **1. Finding the dimensions of the rectangle:** The points \( A(-5, 5) \) and \( B(1, 5) \) lie on the same horizontal line because they share a \( y \)-coordinate of \( 5 \). The length of side \( AB \) is calculated using the distance formula for points along the same horizontal line: \[ \text{Length of } AB = |x_B - x_A| = |1 - (-5)| = |1 + 5| = |6| = 6 \text{ units} \] Next, we know the area of rectangle \( ABCD \) is given as \( 18 \) square units. The area of a rectangle is calculated by the formula: \[ \text{Area} = \text{Length} \times \text{Width} \] \[ 18 = 6 \times \text{Width} \] To find the width, we rearrange this equation: \[ \text{Width} = \frac{18}{6} = 3 \text{ units} \] Thus, the dimensions of the rectangle are: - Length = \( 6 \) units - Width = \( 3 \) units **2. Finding the remaining two points \( C \) and \( D \):** Since \( AB \) is a horizontal side, the remaining two points \( C \) and \( D \) will be vertically aligned with \( A \) and \( B \). Since the width is \( 3 \) units, we will add and subtract \( 3 \) from the \( y \)-coordinates of points \( A \) and \( B \). We can calculate the coordinates: - For point \( A(-5, 5) \), the possible coordinates for point \( D \): - \( D(-5, 5 - 3) \) or \( D(-5, 2) \) - For point \( B(1, 5) \), the possible coordinates for point \( C \): - \( C(1, 5 - 3) \) or \( C(1, 2) \) Thus, the remaining two points that form the rectangle are: - \( C(1, 2) \) and \( D(-5, 2) \). In summary, the dimensions of the rectangle are \( 6 \) units length and \( 3 \) units width, and the coordinates of the points are: - \( A(-5, 5) \) - \( B(1, 5) \) - \( C(1, 2) \) - \( D(-5, 2) \).

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