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Chapter 10: Problem 15

Are these points collinear? a) \(A(-2,5), B(0,2),\) and \(C(4,-4)\) b) \(\quad D(-1,-1), E(2,-2),\) and \(F(5,-5)\)

Short Answer

Expert verified
Both sets of points are collinear as both triangles have an area of 0.

Step by step solution

01

Understand the Concept of Collinearity

Collinear points are points that lie on the same straight line. To check if given points are collinear, we can calculate the area of the triangle formed by the points. If the area is zero, the points are collinear.
02

Verify Collinearity for Points A, B, and C

Calculate the area of triangle formed by points A(-2,5), B(0,2), and C(4,-4) using the formula: \[ \text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|. \] Substitute the coordinates into the formula: \[ \text{Area} = \frac{1}{2} \left| -2(2-(-4)) + 0(-4-5) + 4(5-2) \right|. \] Simplifying, we get: \[ \text{Area} = \frac{1}{2} \left| -12 + 0 + 12 \right| = \frac{1}{2} \times 0 = 0. \] Since the area is 0, points A, B, and C are collinear.
03

Verify Collinearity for Points D, E, and F

Calculate the area of triangle formed by points D(-1,-1), E(2,-2), and F(5,-5) using the same formula: \[ \text{Area} = \frac{1}{2} \left| -1(-2-(-5)) + 2(-5-(-1)) + 5(-1-(-2)) \right|. \] Simplifying, we find: \[ \text{Area} = \frac{1}{2} \left| 3(1) + 2(-4) + 5(1) \right| = \frac{1}{2} \left| 3 - 8 + 5 \right| = \frac{1}{2} \times 0 = 0. \] Since the area is 0, points D, E, and F are collinear.

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Key Concepts

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

geometry
Geometry is a branch of mathematics that deals with shapes, sizes, and the properties of space. In this case, we focus on collinear points, which are points that lie on the same straight line. Ensuring you understand geometry requires a clear grasp of terms and properties. Here, collinearity is checked using the area of a triangle formed by three points. If the area calculation results in zero, it confirms the points are collinear because a triangle with zero area implies the points lie on a straight line.
  • A solid understanding of basic geometry - such as shapes, angles, and lines - is crucial.
  • Visualizing geometric concepts helps in understanding real-world applications.
  • Being comfortable with formulas and calculations can assist in solving geometric problems efficiently.
coordinate plane
Understanding the coordinate plane is essential when working with problems in geometry that involve distances and areas. A coordinate plane is a two-dimensional surface with a horizontal axis (x-axis) and a vertical axis (y-axis). Points on this plane are identified using ordered pairs \((x, y)\), which detail their exact position.
  • The x-coordinate indicates how far to move on the horizontal axis.
  • The y-coordinate tells how far to move on the vertical axis.
  • Every point can be precisely located, which is critical for calculating distances, slopes, and areas.

By plotting points on the coordinate plane, we can determine if they are collinear by calculating the area of the triangle they form, as explained in geometry.
area of a triangle
The area of a triangle is a key concept used to determine if points are collinear in a coordinate plane. The general formula to find the area of a triangle formed by three points \( (x_1, y_1), (x_2, y_2), (x_3, y_3) \) on the coordinate plane is:
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|. \]
To use this formula effectively, substitute the correct x and y values of the points you have.
  • If the calculated area is zero, the points are collinear.
  • The formula effectively uses the properties of the coordinate plane to derive a numerical value representing the area.
  • No need to visualize or draw the triangle; the formula provides a straightforward mathematical solution.

Practicing with this formula strengthens the understanding of how algebra and geometry intersect in solving problems on the coordinate plane.

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Most popular questions from this chapter

Use slopes to decide whether the triangle with vertices at \((6,5),(-3,0),\) and \((4,-2)\) is a right triangle.

The planes with the equations \(x+2 z=12\) and \(y-3 z=6\) intersect in a line. Find the equation for the line in the form \((x, y, z)=\left(x_{1}, y_{1}, z_{1}\right)+n(a, b, c)\)

Determine the point of intersection, if such a point exists, for the line \(\ell:(x, y, z)=(-1,-6,5)+n(1,3,-2)\) and the plane \(2 x+3 y+z=48\)

Draw and label a well-placed figure in the coordinate system for each theorem. Do not attempt to prove the theorem! The line segment joining the midpoints of the two nonparallel sides of a trapezoid is parallel to each base of the trapezoid.

Explain why the lines below intersect. \(\ell_{1}:(x, y, z)=(2,3,4)+n(1,2,-3)\) and \(\ell_{2}:(x, y, z)=(2,3,4)+r(2,3,5)\)
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