GED Math : GED Math

Study concepts, example questions & explanations for GED Math

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Example Questions

Example Question #3 : Midpoint Formula

You are given points \(\displaystyle A (8, 1)\) and \(\displaystyle D (1,9)\). \(\displaystyle C\) is the midpoint of \(\displaystyle \overline{AD}\)\(\displaystyle B\) is the midpoint of \(\displaystyle \overline{AC}\), and \(\displaystyle E\) is the midpoint of \(\displaystyle \overline{BD}\). Give the coordinates of \(\displaystyle E\).

Possible Answers:

\(\displaystyle (5.375, 4)\)

\(\displaystyle (2.125,8)\)

\(\displaystyle (3.625, 6)\)

\(\displaystyle \left ( 7.625, 2\right )\)

Correct answer:

\(\displaystyle (3.625, 6)\)

Explanation:

Repeated application of the midpoint formula, \(\displaystyle \left (\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2} \right )\), yields the following:

\(\displaystyle A\) is the point \(\displaystyle (8,1)\) and \(\displaystyle D\) is the point \(\displaystyle \left (1, 9 \right )\)\(\displaystyle C\) is the midpoint of \(\displaystyle \overline{AD}\), so \(\displaystyle C\) has coordinates

\(\displaystyle \left (\frac{8+1}{2}, \frac{1+9}{2} \right )\), or \(\displaystyle \left ( 4.5, 5\right )\).

 \(\displaystyle B\) is the midpoint of \(\displaystyle \overline{AC}\), so \(\displaystyle B\) has coordinates

\(\displaystyle \left (\frac{8+4.5}{2}, \frac{1+5}{2} \right )\), or \(\displaystyle (6.25, 3)\).

\(\displaystyle E\) is the midpoint of \(\displaystyle \overline{BD}\), so \(\displaystyle E\) has coordinates

\(\displaystyle \left (\frac{6.25+1}{2}, \frac{3+9}{2} \right )\), or \(\displaystyle (3.625, 6)\).

Example Question #4 : Midpoint Formula

What is the midpoint between \(\displaystyle (2,6)\) and \(\displaystyle (1 ,-9)\)?

Possible Answers:

\(\displaystyle (\frac{1}{2}, -\frac{15}{2})\)

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

\(\displaystyle (\frac{3}{2}, -\frac{15}{2})\)

\(\displaystyle (-\frac{1}{2}, -\frac{15}{2})\)

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

Correct answer:

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

Explanation:

Write the formula to find the midpoint.

\(\displaystyle M = (\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})\)

Substitute the points into the equation.

\(\displaystyle M = (\frac{2+1}{2}, \frac{6+(-9)}{2})\)

The midpoint is located at:  \(\displaystyle (\frac{3}{2}, -\frac{3}{2})\)

The answer is:  \(\displaystyle (\frac{3}{2}, -\frac{3}{2})\)

Example Question #4 : Midpoint Formula

Find the midpoint of \(\displaystyle (1,7)\) and \(\displaystyle (-2,-6)\).

Possible Answers:

\(\displaystyle (-\frac{3}{2}, \frac{1}{2})\)

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

\(\displaystyle (3,13)\)

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

\(\displaystyle (-\frac{3}{2}, \frac{13}{2})\)

Correct answer:

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

Explanation:

Write the formula for the midpoint.

\(\displaystyle M=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})\)

Substitute the points.

\(\displaystyle M=(\frac{1+(-2)}{2},\frac{7+(-6)}{2}) = (\frac{-1}{2}, \frac{1}{2})\)

The answer is:  \(\displaystyle (-\frac{1}{2}, \frac{1}{2})\)

Example Question #5 : Midpoint Formula

What is the midpoint between \(\displaystyle (4,5)\) and \(\displaystyle (-7,10)\)?

Possible Answers:

\(\displaystyle (-1.5,7.5)\)

\(\displaystyle (5.5,15)\)

\(\displaystyle (11,15)\)

\(\displaystyle (-3,15)\)

\(\displaystyle (5.5,7.5)\)

Correct answer:

\(\displaystyle (-1.5,7.5)\)

Explanation:

Recall that the general formula for the midpoint between two points is:

\(\displaystyle (\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})\)

Think of this like being the "average" of your two points.

Based on your data, you know that your midpoint could be calculated as follows:

\(\displaystyle (\frac{4-7}{2},\frac{5+10}{2})=(\frac{-3}{2},\frac{15}{2})\)

This is the same as:

\(\displaystyle (-1.5,7.5)\)

Example Question #4 : Midpoint Formula

What is the midpoint between the points \(\displaystyle (-10,4)\) and \(\displaystyle (-22,-4)\)?

Possible Answers:

\(\displaystyle (-16,0)\)

\(\displaystyle (16,8)\)

\(\displaystyle (-32,0)\)

\(\displaystyle (-32,-8)\)

\(\displaystyle (-16,-4)\)

Correct answer:

\(\displaystyle (-16,0)\)

Explanation:

Recall that the general formula for the midpoint between two points is:

\(\displaystyle (\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})\)

Think of this like being the "average" of your two points.

Based on your data, you know that your midpoint could be calculated as follows:

\(\displaystyle (\frac{-10-22}{2},\frac{4-4}{2})=(\frac{-32}{2},\frac{0}{2})\)

This is the same as:

\(\displaystyle (-16,0)\)

Example Question #5 : Midpoint Formula

\(\displaystyle (0,5)\) is the midpoint between \(\displaystyle (84,1)\) and some other point.  What is that point?

Possible Answers:

\(\displaystyle (-42,4)\)

\(\displaystyle (84,10)\)

\(\displaystyle (-84,9)\)

\(\displaystyle (42,4)\)

\(\displaystyle (-84,10)\)

Correct answer:

\(\displaystyle (-84,9)\)

Explanation:

Recall that the general formula for the midpoint between two points is:

\(\displaystyle (\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})\)

Think of this like being the "average" of your two points.

Now, you can write your data out as follows, as you know the midpoint value as well as one of the values for your end points:

\(\displaystyle (0,5)=(\frac{84+x_2}{2},\frac{1+y_2}{2})\)

To finish solving, think of it like two different equations:

\(\displaystyle \frac{84+x_2}{2}=0\) 

and

\(\displaystyle \frac{1+y_2}{2}=5\)

Now, solve each for the respective values:

\(\displaystyle 84+x_2=0\)

\(\displaystyle x_2=-84\)

and

\(\displaystyle 1+y_2=10\)

\(\displaystyle y_2=9\)

Therefore, your other point is \(\displaystyle (x_2,y_2)\) or \(\displaystyle (-84,9)\)

 

Example Question #2 : Midpoint Formula

\(\displaystyle (-10,-22)\) is the midpoint between \(\displaystyle (-2,-7)\) and some other point.  What is that point?

 

Possible Answers:

\(\displaystyle (-18,-37)\)

\(\displaystyle (-12,-29)\)

\(\displaystyle (-16,-25)\)

\(\displaystyle (-18,-44)\)

\(\displaystyle (-5,55)\)

Correct answer:

\(\displaystyle (-18,-37)\)

Explanation:

Recall that the general formula for the midpoint between two points is:

\(\displaystyle (\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})\)

Think of this like being the "average" of your two points.

Now, you can write your data out as follows, as you know the midpoint value as well as one of the values for your end points:

\(\displaystyle (-10,-22)=(\frac{-2+x_2}{2},\frac{-7+y_2}{2})\)

To finish solving, think of it like two different equations:

\(\displaystyle \frac{-2+x_2}{2}=-10\) 

and

\(\displaystyle \frac{-7+y_2}{2}=-22\)

Now, solve each for the respective values:

\(\displaystyle -2+x_2=-20\)

\(\displaystyle x_2=-18\)

and

\(\displaystyle -7+y_2=-44\)

\(\displaystyle y_2=-37\)

Therefore, your other point is \(\displaystyle (x_2,y_2)\) or \(\displaystyle (-18,-37)\)

Example Question #7 : Midpoint Formula

Find the midpoint of the following points:

\(\displaystyle (8,2)\)

\(\displaystyle (10,4)\)

Possible Answers:

\(\displaystyle (9,4)\)

\(\displaystyle (18,6)\)

\(\displaystyle (10,3)\)

\(\displaystyle (2,2)\)

\(\displaystyle (9,3)\)

Correct answer:

\(\displaystyle (9,3)\)

Explanation:

To find the midpoint between two points, we will use the following formula:

\(\displaystyle M = (\frac{x_1 +x_2}{2}, \frac{y_1 +y_2}{2})\)

where \(\displaystyle (x_1,x_2)\) and \(\displaystyle (y_1,y_2)\) are the given points.

So, given the points

\(\displaystyle (8,2)\) and \(\displaystyle (10,4)\), we can substitute into the formula. We get

\(\displaystyle M = (\frac{8+10}{2}, \frac{2+4}{2})\)

\(\displaystyle M = (\frac{18}{2}, \frac{6}{2})\)

\(\displaystyle M = (9,3)\)

Example Question #11 : Midpoint Formula

Find the midpoint given the following points:

\(\displaystyle (9,1)\) and \(\displaystyle (1, 2)\)

Possible Answers:

\(\displaystyle (8, 1.5)\)

\(\displaystyle (8,1)\)

\(\displaystyle (10,3)\)

\(\displaystyle (5,2)\)

\(\displaystyle (5, 1.5)\)

Correct answer:

\(\displaystyle (5, 1.5)\)

Explanation:

To find the midpoint between two points, we will use the following formula:

\(\displaystyle M = (\frac{x_1 +x_2}{2}), (\frac{y_1 +y_2}{2})\)

where \(\displaystyle (x_1,x_2)\) and \(\displaystyle (y_1,y_2)\) are the given points.

So, given the points \(\displaystyle (9,1)\) and \(\displaystyle (1, 2)\), we can substitute. We get

\(\displaystyle M = (\frac{9+1}{2}, \frac{1+2}{2})\)

\(\displaystyle M = (\frac{10}{2}, \frac{3}{2})\)

\(\displaystyle M = (5, 1.5)\)

Example Question #111 : Coordinate Geometry

The midpoint of a line with endpoints at \(\displaystyle (3, 5)\) and \(\displaystyle (-5, y)\) is \(\displaystyle (-2, 12)\). Find the value of \(\displaystyle y\).

Possible Answers:

\(\displaystyle 21\)

\(\displaystyle 17\)

\(\displaystyle 19\)

\(\displaystyle 15\)

Correct answer:

\(\displaystyle 19\)

Explanation:

Recall how to find the midpoint of a line:

\(\displaystyle \text{Midpoint}=(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})\)

Since we are only worried about the \(\displaystyle y\)-coordinate, we can write the following equation:

\(\displaystyle \frac{5+y}{2}=12\)

Solve for \(\displaystyle y\).

\(\displaystyle 5+y=24\)

\(\displaystyle y=19\)

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