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Chapter 13: Problem 9

Find the length, slope, and midpoint of \(\overline{P Q}\). $$P(-7,11), Q(1,-4)$$

Short Answer

Expert verified
The length of \(\overline{P Q}\) is 17 units, the slope of \(\overline{P Q}\) is -1.875, and the midpoint is (-3, 3.5).

Step by step solution

01

STEP 1: Length Calculation

Use the Euclidean distance formula: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]To find the length of the line \(\overline{P Q}\), plug P(-7,11) and Q(1,-4) into the formula: \[d = \sqrt{(1 - (-7))^2 + ((-4) - 11)^2} =\sqrt{(1 + 7)^2 + ((-4) - 11)^2}= \sqrt{(8)^2 + (-15)^2} = \sqrt{64 + 225} = \sqrt{289}=17 \]
02

STEP 2: Slope Calculation

The formula for slope is \[m = \frac{(y_2 - y_1)}{(x_2 - x_1)}.\]Plug coordinates of points P(-7,11) and Q(1,-4) into the formula: \[m = \frac{(-4 - 11)}{(1 - {-7})} = \frac{-15}{8}= -\frac{15}{8}.\]
03

STEP 3: Midpoint Calculation

The formula for the midpoint is \[(\frac{(x_1 + x_2)}{2}, \frac{(y_1 + y_2)}{2}).\]Plug coordinates of points P(-7,11) and Q(1,-4) into the formula: \[(\frac{(-7+1)}{2}, \frac{(11-4)}{2}) = (-3, \frac{7}{2}) = (-3, 3.5).\]

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Key Concepts

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

Euclidean Distance Formula
Understanding how to determine the length of a line segment is crucial in geometry, and the Euclidean distance formula is the tool for this job. This formula, represented as \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\], allows us to calculate the straight-line distance between two points in a coordinate plane. When working with this formula, each point has two coordinates: \(x, y\), where \(x\) represents the horizontal position and \(y\) the vertical position.

To apply this formula properly, follow these steps:
  • Identify the coordinates for both points. For example, if we have points \(P(-7, 11)\) and \(Q(1, -4)\), \(x_1 = -7\), \(y_1 = 11\), \(x_2 = 1\), and \(y_2 = -4\).
  • Plug the coordinates into the formula and simplify. In our example, after simplifications, the distance \(d\) turns out to be 17 units.
The result represents the unit measure distance on the plane, providing an exact length between the two points, thus displaying how far apart they are from each other.
Slope Calculation
The concept of slope is fundamental in understanding the steepness and direction of a line. It's defined as the ratio of the vertical change to the horizontal change between two points on a line, symbolized by \(m\). The slope formula is expressed as \[m = \frac{(y_2 - y_1)}{(x_2 - x_1)}.\]

To find a line's slope, you can follow these steps:
  • Taking our points \(P(-7, 11)\) and \(Q(1, -4)\), subtract the \(y\)-coordinate of \(P\) from the \(y\)-coordinate of \(Q\) to find the numerator of our slope fraction.
  • Subtract the \(x\)-coordinate of \(P\) from the \(x\)-coordinate of \(Q\) for the denominator.
  • Calculating these differences and dividing the former by the latter gives us the slope \(m\), which in this case is \( -\frac{15}{8}\) or \( -1.875\).
This negative value indicates that the line is descending from left to right, and the magnitude of this number tells us that for every 8 units it moves horizontally, it drops 15 units vertically.
Midpoint of a Line Segment
The midpoint of a line segment is the point that divides it into two equal parts. It is quite literally the 'middle' of the segment. Finding this midpoint involves a simple average of the x-coordinates and y-coordinates of the end points. The formula for the midpoint is \[\left(\frac{(x_1 + x_2)}{2}, \frac{(y_1 + y_2)}{2}\right).\]

To use this formula, just do the following:
  • For points \(P(-7, 11)\) and \(Q(1, -4)\), add together the \(x\)-coordinates and then divide by 2 to find the \(x\)-coordinate of the midpoint.
  • Repeat the process with the \(y\)-coordinates to find the midpoint's \(y\)-coordinate.
  • Once we add and average the coordinates accordingly, we determine that the midpoint of \(\overline{P Q}\) is located at \( (-3, 3.5) \).
The midpoint is often used to split line segments for further geometric constructions or in proofs where the bisector might be of interest.

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Most popular questions from this chapter

Given: Points \(N(-1,-5), O(0,0), P(3,2),\) and \(Q(8,1)\) a. Show that \(N O P Q\) is an isosceles trapezoid. b. Show that the diagonals are congruent.

Find the center and the radius of the circle \(x^{2}+4 x+y^{2}-8 y=16\) (Hint: Express the given equation in the form $$ (x-a)^{2}+(y-b)^{2}=r^{2} $$

the line through \((5,5)\) that makes a \(45^{\circ}\) angle measured counterclockwise from the positive \(x\) -axis

Solve each pair of equations algebraically. Then draw the graphs of the equations and label their intersection point. $$\begin{aligned}&3 x+2 y=8\\\&-x+3 y=12\end{aligned}$$

Find the center and the radius of each circle. $$(x-j)^{2}+(y+14)^{2}=17$$
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