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A triangle is graphed on the coordinate plane. Two of its vertices are located at (-10,3) ar (0,-20). The third vertex is located at (2, y), where y is a positive value. The area of the triangle is 158 square units. Using the shoelace formula in counterclockwise fashion, determine the value of y.

A triangle is graphed on the coordinate plane. Two of its vertices are located at (-10,3) ar (0,-20). The third vertex is located at (2, y), where y is a positive value. The area of the triangle is 158 square units. Using the shoelace formula in counterclockwise fashion, determine the value of y.

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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**Title: Calculating the Area of a Triangle Using the Shoelace Formula** **Problem Statement:** A triangle is graphed on the coordinate plane. Two of its vertices are located at \((-10, 3)\) and \((0, -20)\). The third vertex is located at \((2, y)\), where \(y\) is a positive value. The area of the triangle is 158 square units. Using the shoelace formula in a counterclockwise fashion, determine the value of \(y\). **Explanation of the Shoelace Formula:** The shoelace formula (also known as Gauss's area formula) is a mathematical algorithm to find the area of a simple polygon when the coordinates of the vertices are known. For a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), \((x_3, y_3)\), and considering the coordinates in counterclockwise order, the formula is given by: \[ \text{Area} = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_1 - y_1x_2 - y_2x_3 - y_3x_1 \right| \] **Application to This Problem:** 1. Assign the vertices as follows: - \((x_1, y_1) = (-10, 3)\) - \((x_2, y_2) = (0, -20)\) - \((x_3, y_3) = (2, y)\) 2. Substitute the coordinates into the Shoelace formula and solve for \(y\). \[ \text{Area} = \frac{1}{2} \left| -10(-20) + 0(y) + 2(3) - (3)(0) - (-20)(2) - y(-10) \right| = 158 \] \[ \frac{1}{2} \left| 200 + 6 + 20 + 10y \right| = 158 \] \[ \frac{1}{2} (226 + 10y) = 158 \] \[ 226 + 10y = 316 \] \[ 10y = 90
Transcribed Image Text:**Title: Calculating the Area of a Triangle Using the Shoelace Formula** **Problem Statement:** A triangle is graphed on the coordinate plane. Two of its vertices are located at \((-10, 3)\) and \((0, -20)\). The third vertex is located at \((2, y)\), where \(y\) is a positive value. The area of the triangle is 158 square units. Using the shoelace formula in a counterclockwise fashion, determine the value of \(y\). **Explanation of the Shoelace Formula:** The shoelace formula (also known as Gauss's area formula) is a mathematical algorithm to find the area of a simple polygon when the coordinates of the vertices are known. For a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), \((x_3, y_3)\), and considering the coordinates in counterclockwise order, the formula is given by: \[ \text{Area} = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_1 - y_1x_2 - y_2x_3 - y_3x_1 \right| \] **Application to This Problem:** 1. Assign the vertices as follows: - \((x_1, y_1) = (-10, 3)\) - \((x_2, y_2) = (0, -20)\) - \((x_3, y_3) = (2, y)\) 2. Substitute the coordinates into the Shoelace formula and solve for \(y\). \[ \text{Area} = \frac{1}{2} \left| -10(-20) + 0(y) + 2(3) - (3)(0) - (-20)(2) - y(-10) \right| = 158 \] \[ \frac{1}{2} \left| 200 + 6 + 20 + 10y \right| = 158 \] \[ \frac{1}{2} (226 + 10y) = 158 \] \[ 226 + 10y = 316 \] \[ 10y = 90
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