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Chapter 0: Problem 8

Calculate the distance between the given points, and find the midpoint of the segment joining them. $$(-4,5) \text { and }(-9,-7)$$

Short Answer

Expert verified
The distance is 13 units and the midpoint is (-6.5, -1).

Step by step solution

01

Use the Distance Formula

To calculate the distance between the two points, use the distance formula, which is: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]Substitute the given points \((x_1, y_1) = (-4, 5)\) and \((x_2, y_2) = (-9, -7)\): \[d = \sqrt{((-9) - (-4))^2 + ((-7) - 5)^2}\]Simplify inside the square root: \[d = \sqrt{(-9 + 4)^2 + (-7 - 5)^2} = \sqrt{(-5)^2 + (-12)^2}\]Now evaluate the expressions: \[d = \sqrt{25 + 144} = \sqrt{169} = 13\]
02

Use the Midpoint Formula

The formula for finding the midpoint \(M\) of a segment connecting two points is: \[M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]Plug in the coordinates \((x_1, y_1) = (-4, 5)\) and \((x_2, y_2) = (-9, -7)\): \[M = \left( \frac{-4 + (-9)}{2}, \frac{5 + (-7)}{2} \right) = \left( \frac{-13}{2}, \frac{-2}{2} \right)\]Simplify each part of the midpoint to get the final point: \[M = \left( -6.5, -1 \right)\]

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Key Concepts

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

Midpoint Formula
Invented to help find the center point between two positions on a plane, the midpoint formula comes in handy whenever you need to split a line segment exactly in half. The formula is super easy to remember: it鈥檚 the average of the x-coordinates and the y-coordinates of your points. When using the midpoint formula, remember:
  • Label your points as \( (x_1, y_1) \) and \( (x_2, y_2) \).
  • Compute the average of the x-coordinates: \[ \frac{x_1 + x_2}{2} \].
  • Compute the average of the y-coordinates: \[ \frac{y_1 + y_2}{2} \].
By plugging in the values, you'll find the coordinates of the midpoint. This makes it useful when determining a point that is exactly in the middle between any two given points.
Coordinate Geometry
Coordinate Geometry is also known as analytic geometry. It鈥檚 a branch of geometry where algebra is used to understand and solve geometric problems. By plotting points, lines, and shapes on a coordinate plane, it becomes much easier to visualize spatial relationships.
In Coordinate Geometry:
  • A point is represented as an ordered pair \( (x, y) \) on the Cartesian plane.
  • Distance and slopes of lines can be calculated using formulas.
  • It serves as a bridge between algebra and geometry, leveraging numerical values to explain geometric concepts.
Understanding how to move within this system is crucial for solving problems like finding distances and midpoints. It opens doors to visualizing complex algebraic equations geometrically.
Distance Calculation
Distance calculation is about determining how far one point is from another on a coordinate plane. The distance formula is your go-to tool for this task. Its applications range from everyday measurement needs like determining the length of a path, to more engineering-focused tasks.
The formula is derived from the Pythagorean theorem, expressing the distance \( d \) as \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \].
Steps to use the formula:
  • Subtract corresponding coordinates: \( x_2 - x_1\) and \( y_2 - y_1 \).
  • Square the differences.
  • Add the squared values.
  • Take the square root of the result.
By following these simple steps, you can confidently determine the distance separating any two points on a plane.
Geometric Formulas
In the world of mathematics, geometric formulas serve as solutions to a myriad of problems. They provide methods for calculation and allow for the conversion of theoretical concepts into practical applications.
Several key geometric formulas exist:
  • Area and perimeter formulas for shapes.
  • The Pythagorean theorem for right triangle calculations.
  • The distance and midpoint formulas for applications in coordinate geometry.
Mastering these formulas is essential as they form the backbone of tasks ranging from simple area calculation to complex architectural designs. Understanding and knowing when to apply these formulas helps solve real-world problems in fields ranging from construction to navigation.

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Most popular questions from this chapter

Find the equation of the line that passes through the given point and also satisfies the additional piece of information. Express your answer in slope- intercept form, if possible. (1,4)\(;\) perpendicular to the line \(-\frac{2}{3} x+\frac{3}{2} y=-2\)

Find the number \(a\) for which \(y=2\) is a solution of the equation \(y-a=y+5-3 a y\)

Write a quadratic equation in standard form whose solution set is \(\\{2-i, 2+i\\} .\) Alternate solutions are possible.

Solve the equation \(x^{2}-2 x=-2\) by first writing in standard form and then using the quadratic formula. Now plot both sides of the equation in the same viewing screen \(\left(y_{1}=x^{2}-2 x \text { and } y_{2}=-2\right) .\) Do these graphs intersect? Does this agree with the solution set you found?

Show that two lines with equal slopes and different \(y\) -intercepts have no point in common. Hint: Let \(y_{1}=m x+b_{1}\) and \(y_{2}=m x+b_{2}\) with \(b_{1} \neq b_{2} .\) What equation must be true for there to be a point of intersection? Show that this leads to a contradiction.
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