Intermediate Geometry : Other Lines

Study concepts, example questions & explanations for Intermediate Geometry

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Example Questions

Example Question #1 : How To Find Out If A Point Is On A Line With An Equation

A line passes through both the origin and the point \displaystyle (12,5). Which of the following points are NOT on the line?

Possible Answers:

\displaystyle (20.4,8.5)

\displaystyle (14.4,6)

\displaystyle (6,2.5)

\displaystyle (-3,-1.25)

\displaystyle (15.6,8)

Correct answer:

\displaystyle (15.6,8)

Explanation:

Because the line passes through the origin, all of the points on the line will have an \displaystyle x:y ratio of \displaystyle 12:5.  Only \displaystyle \left ( 15.6,8 \right ) does not meet this requirement.

Example Question #1 : How To Find Out If A Point Is On A Line With An Equation

Which of the following points does NOT lie on the graph of \displaystyle y=x^2-3x+7

Possible Answers:

\displaystyle (5,17)

\displaystyle (1,5)

\displaystyle (4,12)

\displaystyle (2,5)

\displaystyle (3,7)

Correct answer:

\displaystyle (4,12)

Explanation:

To find if a point lies on a graph or not, simply plug the x and y values into your equation and see if it holds true. Plugging our x values into the equation \displaystyle y=x^2-3x+7 gives us the following:

 

\displaystyle x=1: y=(1)^2-3(1)+7=5

\displaystyle x=2: y=(2)^2-3(2)+7=5

\displaystyle x=3: y=(3)^2-3(3)+7=7

\displaystyle x=4: y=(4)^2-3(4)+7=11 and

\displaystyle x=5: y=(5)^2-3(5)+7=17

 

So our points are \displaystyle (1,5), (2,5), (3,7), (4,11),and \displaystyle (5,17). This makes \displaystyle (4,12) the only point that does not lie on our graph. 

Example Question #1 : How To Find Out If A Point Is On A Line With An Equation

Which of the following points exists on the line \displaystyle 2x-2y=1?

Possible Answers:

\displaystyle (0,1)

\displaystyle (1,0)

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

\displaystyle (0,\frac{1}{2})

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

Correct answer:

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

Explanation:

Substitute each answer choice into the equation in question, \displaystyle 2x-2y=1, in order to test if the equation is valid at the given point

Only \displaystyle ( 1,\frac{1}{2}) will satisfy the equation on both the left and right side.

Example Question #1 : Other Lines

Which of the following points exist on the line \displaystyle x=1?

Possible Answers:

\displaystyle (0,1)

\displaystyle (1,20)

\displaystyle (20,1)

\displaystyle (0,-1)

\displaystyle (-1,0)

Correct answer:

\displaystyle (1,20)

Explanation:

The line \displaystyle x=1 is a vertical line. This means that any point with an \displaystyle x-value of \displaystyle 1 will exist on the line.

The only possible point with an \displaystyle x-value of \displaystyle 1 is \displaystyle (1,20).

Example Question #5 : How To Find Out If A Point Is On A Line With An Equation

Which point will exist on the line \displaystyle y=-1?

Possible Answers:

\displaystyle (-1,1)

\displaystyle (1000,-1)

\displaystyle (-1,0)

\displaystyle (-1,1000)

\displaystyle (1000,1000)

Correct answer:

\displaystyle (1000,-1)

Explanation:

The line \displaystyle y=-1 is a horizontal line with points all \displaystyle y-values equal to \displaystyle 1.

The only possible answer with a \displaystyle y-value of \displaystyle -1 is \displaystyle (1000,-1).

Example Question #2 : Other Lines

Which point is on the line \displaystyle y = 2x - 4 ?

Possible Answers:

\displaystyle (1,-2)

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

\displaystyle (-1,-6)

\displaystyle (2,0)

\displaystyle (-\frac{1}{2}, 5)

Correct answer:

\displaystyle (-\frac{1}{2}, 5)

Explanation:

A point is on a line if, plugged in, it creates a true mathematical equation.

For example, \displaystyle (1,-2) is definitely on the line because when you plug it in, it creates a true equation:

\displaystyle y = 2x - 4 plug in the \displaystyle x and \displaystyle y values

\displaystyle -2 = 2(1) - 4 multiply

\displaystyle -2 = 2-4

\displaystyle -2 = -2 this is true, so that point is on the line

The point that does not work is \displaystyle (-\frac{1}{2}, 5):

\displaystyle 5 = 2(-\frac{1}{2 }) -4 multiply

\displaystyle 5 = -1 - 4 subtract

\displaystyle 5 \neq -5

Since this is not true, the point is NOT on the line.

Example Question #3 : Other Lines

Which point is NOT on the line \displaystyle y = \frac{2}{3} x - 5?

Possible Answers:

\displaystyle (6,-9)

\displaystyle (3,-3)

\displaystyle (-3, -7)

\displaystyle (6, -1 )

\displaystyle (6, 1)

Correct answer:

\displaystyle (6, 1)

Explanation:

A point is on a line if, plugged in, it creates a true mathematical equation.

For example, \displaystyle (3,-3) is definitely on the line because when you plug it in, it creates a true equation:

\displaystyle y = \frac{2}{3}x - 5 plug in the \displaystyle x and \displaystyle y values

\displaystyle -3 =\frac{2}{3}(3) - 5 multiply

\displaystyle -3 = 2-5

\displaystyle -3= -3 this is true, so that point is on the line

The point that does not work is \displaystyle (6, 1):

\displaystyle 1 = \frac{2}{3}(6) -5 multiply

\displaystyle 1 = 4- 5 subtract

\displaystyle 1 \neq -1

Since this is not true, the point is NOT on the line.

Example Question #4 : Other Lines

Which point is NOT on the line \displaystyle y = \frac{1}{4} x + 3 ?

Possible Answers:

\displaystyle (-8, 1)

\displaystyle (8,5)

\displaystyle (-4,4)

\displaystyle (-4,2)

\displaystyle (4,4)

Correct answer:

\displaystyle (-4,4)

Explanation:

A point is on a line if, plugged in, it creates a true mathematical equation.

For example, \displaystyle (-4, 2) is definitely on the line because when you plug it in, it creates a true equation:

\displaystyle y = \frac{1}{4}x +3 plug in the \displaystyle x and \displaystyle y values

\displaystyle 2 = \frac{1}{4}(-4) +3 multiply

\displaystyle 2 = -1+3

\displaystyle 2 =2 this is true, so that point is on the line

The point that does not work is \displaystyle (-4, 4):

\displaystyle 4 =\frac{1}{4} (-4)+3 multiply

\displaystyle 4 = -1 +3 subtract

\displaystyle 4 \neq -2

Since this is not true, the point is NOT on the line.

Example Question #5 : Other Lines

Which point is NOT on the line \displaystyle y = 3x+1 ?

Possible Answers:

\displaystyle (-1,-4)

\displaystyle (-3,-8)

\displaystyle (3,10)

\displaystyle (-1,-2)

\displaystyle (1,4)

Correct answer:

\displaystyle (-1,-4)

Explanation:

A point is on a line if, plugged in, it creates a true mathematical equation.

For example, \displaystyle (-1,-2) is definitely on the line because when you plug it in, it creates a true equation:

\displaystyle y = 3x+1 plug in the \displaystyle x and \displaystyle y values

\displaystyle -2 = 3(-1)+1 multiply

\displaystyle -2 = -3+1

\displaystyle -2 = -2 this is true, so that point is on the line

The point that does not work is \displaystyle (-1, -4):

\displaystyle -4=3(-1)+1 multiply

\displaystyle -4 = -3+1 subtract

\displaystyle -4 \neq -2

Since this is not true, the point is NOT on the line.

Example Question #6 : Other Lines

The equation of a line is given below:

\displaystyle y=\frac{1}{4}x + 2

which point below can be found is NOT on the line?

Possible Answers:

\displaystyle (0,2)

\displaystyle (8,2)

\displaystyle (2,2.5)

\displaystyle (4,3)

\displaystyle (-8,0)

Correct answer:

\displaystyle (8,2)

Explanation:

To find the solution to this problem just do a quick check, pluggin in each point's x and y value into the equation.  Remember coordinates are written with the x-value first, then the y-value (x,y).

If you try each point the only one that doesn't work is (8,2) !

\displaystyle y=\frac{1}{4} x + 2 = \frac{1}{4}(8) +2 = \frac{8}{4} +2 = 2 + 2 = 4 \neq 2

In other words, if the x-value is 8 the y-value should be 4!

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