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Chapter 6: Problem 48

\(A=(-4,-5), B=(3,10), C=(6,12)\)

Short Answer

Expert verified
The distance between points A and B is approximately 16.16 units, between points A and C is approximately 19.72 units, and between points B and C is approximately 3.61 units.

Step by step solution

01

Calculate the distance between points A and B

To calculate the distance between points A and B, apply the distance formula where \(x_1 = -4\), \(y_1 = -5\), \(x_2 = 3\), and \(y_2 = 10\): \[AB = \sqrt{(3 - (-4))^2 + (10 - (-5))^2}\]
02

Calculate the distance between points A and C

To calculate the distance between points A and C, apply the distance formula where \(x_1 = -4\), \(y_1 = -5\), \(x_2 = 6\), and \(y_2 = 12\): \[AC = \sqrt{(6 - (-4))^2 + (12 - (-5))^2}\]
03

Calculate the distance between points B and C

To calculate the distance between points B and C, apply the distance formula where \(x_1 = 3\), \(y_1 = 10\), \(x_2 = 6\), and \(y_2 = 12\): \[BC = \sqrt{(6 - 3)^2 + (12 - 10)^2}\]

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Key Concepts

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

Coordinate Geometry
Coordinate Geometry is a branch of geometry where the position of points on a plane is described using an ordered pair of numbers. This system, known as the Cartesian coordinate system, enables us to use algebraic techniques to solve geometric problems. In our exercise, the points A, B, and C each have pairs of coordinates that show their position on the coordinate plane.
  • The first number in each pair, called the x-coordinate, tells how far to move along the horizontal axis, or x-axis.
  • The second number, the y-coordinate, indicates how far to move along the vertical, or y-axis.
This way of representing points is very useful for performing calculations like finding the distance between two points, which is a common application of Coordinate Geometry.
Calculating Distances
The Distance Formula is a tool derived from the Pythagorean theorem, used in geometry to find the length between two points on a coordinate plane. It looks like this: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]In this formula:
  • \( (x_1, y_1) \) and \( (x_2, y_2) \) represent the coordinates of the two points.
  • The differences \(x_2 - x_1\) and \(y_2 - y_1\) measure how far apart the points are horizontally and vertically.
  • Squaring these differences eliminates negatives and ensures we only deal with distances, not direction.
  • Finally, taking the square root gives us the true distance between the two points.
Using the Distance Formula, you can easily calculate how far apart any two points are, as demonstrated in the exercise with points A, B, and C.
Step-by-Step Solutions
Breaking down a math problem into manageable steps is a great strategy for understanding and solving it. Consider each phase in our exercise:
  • Step 1: We calculated the distance between points A and B. Identifying each coordinate and substituting these values into the Distance Formula helped us find the length of segment AB.
  • Step 2: Likewise, for calculating the distance between points A and C, we followed the same process, plugging the corresponding coordinates into the formula. This step-by-step repetition reinforces the concept.
  • Step 3: Finally, measuring the distance between points B and C provided further practice and confidence in applying the Distance Formula efficiently.
By taking each piece of the problem and dealing with it individually, we make complex tasks more approachable. This step-by-step approach is invaluable, especially when learning new concepts in mathematics.

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Most popular questions from this chapter

In Exercises 33-48, find a polar equation of the conic with its focus at the pole. $$ \begin{array}{lll} {\text { Conic }} & \text { Eccentricity } & \text { Directrix } \\ \text { Hyperbola } & e=2 & x=1 \\ \end{array} $$

Show that the polar equation of the ellipse $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \quad \text { is } \quad r^{2}=\frac{b^{2}}{1-e^{2} \cos ^{2} \theta} $$

In Exercises 33-48, find a polar equation of the conic with its focus at the pole. $$ \begin{array}{ll} {\text { Conic }} & \text { Vertex or Vertices } \\ \text { Parabola } & (1,-\pi / 2) \\ \end{array} $$

In Exercises 73-78, solve the trigonometric equation. $$ 9 \csc ^{2} x-10=2 $$

\(r=2 \sec \theta\)
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