HiSET: Math : Midpoint of line segments

Study concepts, example questions & explanations for HiSET: Math

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Example Questions

Example Question #1 : Midpoint Of Line Segments

Give the coordinates of the midpoint of the line segment whose endpoints are the intercepts of the line of the equation

\(\displaystyle 2x+ 5y = 40\).

Possible Answers:

\(\displaystyle (-14, -4)\)

None of the other choices gives the correct response.

\(\displaystyle (4, 14)\)

\(\displaystyle (14, 4)\)

\(\displaystyle (-4, -14)\)

Correct answer:

\(\displaystyle (14, 4)\)

Explanation:

The x-intercept and y-intercept of the line of an equation can be found by substituting 0 for \(\displaystyle y\) and \(\displaystyle x\), respectively, as follows:

x-intercept:

\(\displaystyle 2x+ 5y = 40\)

Substitute and simplify:

\(\displaystyle 2x+ 5(0)= 40\)

\(\displaystyle 2x+ 0 = 40\)

\(\displaystyle 2x= 40\)

Solve for \(\displaystyle x\) by dividing by 2 on both sides:

\(\displaystyle 2x \div 2 = 40 \div 2\)

\(\displaystyle x = 20\)

The x-intercept of the line is located at \(\displaystyle (20,0)\)

The y-intercept can be found by setting \(\displaystyle x= 0\) and solving for \(\displaystyle y\) in a similar fashion:

\(\displaystyle 2x+ 5y = 40\)

\(\displaystyle 2 (0)+ 5y = 40\)

\(\displaystyle 0 + 5y = 40\)

\(\displaystyle 5y = 40\)

\(\displaystyle 5y \div 5 = 40 \div 5\)

\(\displaystyle y = 8\)

The y-intercept is located at \(\displaystyle (0,8)\).

 

The midpoint of a line segment, given its endpoints \(\displaystyle (x_{1}, y_{1})\) and \(\displaystyle (x_{2}, y_{2})\), is located at

\(\displaystyle \left ( \frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2} \right )\);

substituting accordingly,

\(\displaystyle \frac{x_{1}+x_{2}}{2} =\frac{20+8}{2} = \frac{28}{2} = 14\)

\(\displaystyle \frac{y_{1}+y_{2}}{2} = \frac{ 0+8}{2} = \frac{8}{2} = 4\)

The midpoint is located at \(\displaystyle (14, 4)\).

 

Example Question #1 : Midpoint Of Line Segments

What is the midpoint of the line segment created by the points \(\displaystyle (-2,16)\) and \(\displaystyle (3,7)\)?

Possible Answers:

\(\displaystyle (\frac{3}{2},\frac{3}{2})\)

\(\displaystyle (-\frac{5}{2},-\frac{13}{2})\)

\(\displaystyle (\frac{1}{2},\frac{23}{2})\)

\(\displaystyle (\frac{9}{2},-\frac{17}{2})\)

\(\displaystyle (\frac{23}{2},\frac{3}{2})\)

Correct answer:

\(\displaystyle (\frac{1}{2},\frac{23}{2})\)

Explanation:

The midpoint of the line segment between two points 

\(\displaystyle (x_{0},y_{0})\) and \(\displaystyle (x_{1},y_{1})\)

is given by the formula

\(\displaystyle (\frac{x_{0}+x_{1}}{2},\frac{y_{0}+y_{1}}{2})\).

Thus, since the line segment is generated by 

\(\displaystyle (-2,16)\) and \(\displaystyle (3,7)\)

applying the midpoint formula we have

 \(\displaystyle (\frac{-2+3}{2},\frac{16+7}{2})= (\frac{1}{2},\frac{23}{2})\)

as the midpoint.

Example Question #1 : Midpoint Of Line Segments

What is the midpoint between \(\displaystyle A(-12,10)\) and \(\displaystyle B(14,-6)\)?

Possible Answers:

\(\displaystyle (-1,2)\)

\(\displaystyle (2,1)\)

\(\displaystyle (-2,-1)\)

\(\displaystyle (1,2)\)

Correct answer:

\(\displaystyle (1,2)\)

Explanation:

Step 1: Recall the midpoint Formula....

\(\displaystyle (\dfrac {x_1+x_2}{2},\dfrac {y_1+y_2}{2})\), where \(\displaystyle (x_1,y_1)\) is the first point and \(\displaystyle (x_2,y_2)\) is the second point

Step 2: Identify the points and their specific terms...

\(\displaystyle x_1=-12, y_1=10, x_2=14, y_2=-6\)

Step 3: Plug in the values into the midpoint formula...

\(\displaystyle (\dfrac {-12+14}{2},\dfrac {10-6}{2})\)

Simplify...

\(\displaystyle (\dfrac {2}{2},\dfrac {4}{2})=(1,2)\)

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