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Chapter 6: Problem 46

Use the concept of the area of a triangle discussed in Exercises \(39-44\) to determine whether the three points are collinear. $$(3,6),(-1,-6),(5,11)$$

Short Answer

Expert verified
The points are not collinear.

Step by step solution

01

Formula for Area of Triangle

To determine if the points \((3,6), (-1,-6), (5,11)\) are collinear, we use the formula for the area of a triangle given three vertices \((x_1, y_1), (x_2, y_2), (x_3, y_3)\): \[A = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|\]If the area \(A\) is zero, then the points are collinear.
02

Substitute the Points into the Formula

Substitute the given points \((3,6), (-1,-6), (5,11)\) into the formula:\[A = \frac{1}{2} \left| 3(-6-11) + (-1)(11-6) + 5(6+6) \right|\]
03

Simplify Inside the Absolute Value

Calculate the values inside the absolute value:\[= \frac{1}{2} \left| 3(-17) + (-1)(5) + 5(12) \right|= \frac{1}{2} \left| -51 - 5 + 60 \right|= \frac{1}{2} \left| 4 \right|\]
04

Calculate the Area

Now compute the area:\[A = \frac{1}{2} \times 4 = 2\]
05

Check for Collinearity

Since the area \(A = 2\) is not zero, the points \((3,6), (-1,-6), (5,11)\) are not collinear.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Area of a Triangle
The area of a triangle is a concept from geometry used to measure the surface covered by the triangle's three sides. To find the area of a triangle when you have its vertices in a coordinate system, we use a specific formula that leverages the coordinates of its vertices.
  • If you're given the three points as coordinates, say \((x_1, y_1), (x_2, y_2), (x_3, y_3)\), the area \(A\) can be calculated using:

    \[A = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|\]
This formula outputs the area covered within the triangle. If the calculated area is zero, it means all three points lie on a single straight line, which implies they are collinear.
Vertices
Vertices are the corners or points where two or more lines meet; in the context of geometry, especially triangles, they are the endpoints of the triangle鈥檚 sides. For a triangle denoted by three points \((x_1, y_1), (x_2, y_2), (x_3, y_3)\), each of these points represents a vertex.
These vertices determine the shape and size of a triangle.
  • In practical problems, knowing the coordinates of vertices allows you to compute other properties, such as side lengths and angles, using the distance formula or trigonometric ratios.
  • When vertices of a triangle are given in a coordinate plane, one can also explore properties such as collinearity for real-world applications.
Absolute Value
Absolute value denotes the magnitude or size of a number, regardless of its sign. It tells us how far a number is from zero on the number line.
In the context of the area of a triangle, we apply it to ensure that the area calculation remains non-negative.
  • For example: If computing the area, and an expression yields a negative result, applying the absolute value \( |x| \) converts it to positive.
  • This is crucial when you calculate algebraic expressions, especially in determinants or geometric computations, to avoid negative interpretations of physical quantities like area.
  • The formula for the area of the triangle deliberately includes absolute value to accommodate any order of vertex insertion as it does not impact the area measurement.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, is a branch of geometry where we use a system of coordinates to define geometric elements and relations.
It allows us to address problems that involve finding distances, midpoints, gradients, and angles using algebra.
  • Using Cartesian coordinates, any geometric shape or line can be described with algebraic equations.
  • It bridges geometry and algebra, allowing the use of graphing and equations to solve geometric problems like areas or collinear checks, as shown with triangles here.
  • In our example, using coordinate geometry, we plug in vertex coordinates directly into a formula to reveal if points align on a straight line, i.e., if they are collinear.

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Most popular questions from this chapter

From January to June \(2012,\) Samsung and Apple spent a combined 293 million dollars on media. Apple spent 93 million dollars more than Samsung. (a) Write a system of equations whose solution gives the spending of each media company, in millions of dollars. Let \(x\) be the amount spent by Apple and \(y\) be the amount spent by Samsung. (b) Solve the system of equations. (c) Interpret the solution.

Let \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right], B=\left[\begin{array}{ll}b_{11} & b_{12} \\ b_{21} & b_{22}\end{array}\right],\) and \(C=\left[\begin{array}{ll}c_{11} & c_{12} \\\ c_{21} & c_{22}\end{array}\right]\) where all the elements are real numbers. Use these matrices to show that each statement is true for \(2 \times 2\) matrices. \(A+B=B+A\) (commutative property)

Solve each system. Write solutions in terms of \(z\) if necessary. $$\begin{aligned} x-2 y+z &=5 \\ -2 x+4 y-2 z &=2 \\ 2 x+y-z &=2 \end{aligned}$$

The average of self-reported spending "yesterday" for high-income consumers and middle-/low-income consumers was 93.50 dollars in September 2012 . High- income consumers spend 65 dollars more than middle-/low-income consumers. (Source: www.marketingcharts.com) (a) Write a system of equations whose solution gives the self-reported spending for each income group. Let \(x\) be the spending by high-income consumers and \(y\) be the spending by middle-/low-income consumers. (b) Solve the system. (c) Interpret the solution.

Solve each system graphically. Give \(x\) - and y-coordinates correct to the nearest hundredth. $$\begin{aligned}y &=5^{x} \\\x y &=1\end{aligned}$$
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