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Chapter 2: Problem 76

In your own words, describe how to find the midpoint of a line segment if its endpoints are known.

Short Answer

Expert verified
The midpoint of a line segment whose endpoints are \((x_1, y_1)\) and \((x_2, y_2)\) is found by using the formulas \((x_1 + x_2) / 2\) and \((y_1 + y_2) / 2\) for the x and y coordinates respectively. So, the midpoint is \((x_m, y_m)\) where \(x_m = (x_1 + x_2) / 2\) and \(y_m = (y_1 + y_2) / 2\).

Step by step solution

01

Identify the coordinates of the end points

Suppose the coordinates of the two endpoints are \((x_1, y_1)\) and \((x_2, y_2)\) respectively. This means that one end of the line segment is at point \((x_1, y_1)\) and the other end is at point \((x_2, y_2)\).
02

Calculate the x-coordinate of the midpoint

The x-coordinate of the midpoint can be found by averaging the x-coordinates of the endpoints. This is done by adding \(x_1\) and \(x_2\) together, and then dividing by 2. This gives the formula \((x_1 + x_2) / 2\). Calculate the result of this formula to get the x-coordinate of the midpoint.
03

Calculate the y-coordinate of the midpoint

Similarly, the y-coordinate of the midpoint can be found by averaging the y-coordinates of the endpoints. This is done by adding \(y_1\) and \(y_2\) together, and then dividing by 2. This gives the formula \((y_1 + y_2) / 2\). Calculate the result of this formula to get the y-coordinate of the midpoint.
04

Determine the coordinates of the midpoint

After getting the x and y coordinates of the midpoint, you can present the coordinates of the midpoint as a pair \((x_m, y_m)\), where \(x_m\) is the result from Step 2 and \(y_m\) is the result from Step 3. The pair \((x_m, y_m)\) constitutes the coordinates of the midpoint of the line segment.

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Most popular questions from this chapter

Graph the given square root functions, \(f\) and \(g,\) in the same rectangular coordinate system. Use the integer values of \(x\) given to the right of each function to obtain ordered pairs. Because only nonnegative numbers have square roots that are real numbers, be sure that each graph appears only for values of \(x\) that cause the expression under the radical sign to be greater than or equal to zero. Once you have obtained your graphs, describe how the graph of g is related to the graph of \(f\). $$ \begin{aligned} &f(x)=\sqrt{x} \quad(x=0,1,4,9) \text { and }\\\ &g(x)=\sqrt{x}+2 \quad(x=0,1,4,9) \end{aligned} $$

Use transformations of the graph of the greatest integer function, \(f(x)=\operatorname{int}(x),\) to graph each function. $$g(x)=2 \operatorname{int}(x+1)$$

write the standard form of the equation of the circle with the given center and radius. $$ \text { Center }(-3,5), r=3 $$

graph both equations in the same rectangular coordinate system and find all points of intersection. Then show that these ordered pairs satisfy the equations. $$ \begin{aligned} x^{2}+y^{2} &=16 \\ x-y &=4 \end{aligned} $$

Will help you prepare for the material covered in the next section. Solve for \(y: \quad x=\frac{5}{y}+4\).
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