Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Can anyone please help me to solve this problem? I am stuck!
![**Problem Statement: Triangle Area Calculation**
Given the vertices of a triangle:
- \( Q(5, 1, 3) \)
- \( R(2, 4, 6) \)
- \( S(-1, 4, 9) \)
Find the area of the triangle.
**Solution Approach: Determinant Method**
To find the area of a triangle in 3D space given the vertices \( (x_1, y_1, z_1), (x_2, y_2, z_2), (x_3, y_3, z_3) \), you can use the following determinant formula:
\[
\text{Area} = \frac{1}{2} \times \sqrt{(y_2z_3 + y_3z_1 + y_1z_2 - y_3z_2 - y_1z_3 - y_2z_1)^2 + (z_2x_3 + z_3x_1 + z_1x_2 - z_3x_2 - z_1x_3 - z_2x_1)^2 + (x_2y_3 + x_3y_1 + x_1y_2 - x_3y_2 - x_1y_3 - x_2y_1)^2}
\]
Let's compute the area using these coordinates.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F99d15f92-0bff-4b4d-a47e-2ac33d144271%2F5754a509-93f1-4847-9743-1b06bba67fb9%2Fpler84u_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement: Triangle Area Calculation**
Given the vertices of a triangle:
- \( Q(5, 1, 3) \)
- \( R(2, 4, 6) \)
- \( S(-1, 4, 9) \)
Find the area of the triangle.
**Solution Approach: Determinant Method**
To find the area of a triangle in 3D space given the vertices \( (x_1, y_1, z_1), (x_2, y_2, z_2), (x_3, y_3, z_3) \), you can use the following determinant formula:
\[
\text{Area} = \frac{1}{2} \times \sqrt{(y_2z_3 + y_3z_1 + y_1z_2 - y_3z_2 - y_1z_3 - y_2z_1)^2 + (z_2x_3 + z_3x_1 + z_1x_2 - z_3x_2 - z_1x_3 - z_2x_1)^2 + (x_2y_3 + x_3y_1 + x_1y_2 - x_3y_2 - x_1y_3 - x_2y_1)^2}
\]
Let's compute the area using these coordinates.
![**Task: Finding the Area of a Triangle**
**Problem Statement:**
Find the area of the triangle with vertices at:
- \( Q(5, 7) \)
- \( R(2, 4) \)
- \( S(-1, 4) \)
[__] (Blank space for calculations or answers)
**Instructions:**
To find the area of a triangle given its vertices on a coordinate plane, you can use the formula:
\[
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
\]
Where \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) are the coordinates of the vertices \(Q\), \(R\), and \(S\).
Plug in the coordinates and perform the calculation to determine the area of the triangle.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F99d15f92-0bff-4b4d-a47e-2ac33d144271%2F5754a509-93f1-4847-9743-1b06bba67fb9%2Fnofvlo9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Task: Finding the Area of a Triangle**
**Problem Statement:**
Find the area of the triangle with vertices at:
- \( Q(5, 7) \)
- \( R(2, 4) \)
- \( S(-1, 4) \)
[__] (Blank space for calculations or answers)
**Instructions:**
To find the area of a triangle given its vertices on a coordinate plane, you can use the formula:
\[
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
\]
Where \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) are the coordinates of the vertices \(Q\), \(R\), and \(S\).
Plug in the coordinates and perform the calculation to determine the area of the triangle.
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