PSAT Math : Points and Distance Formula

Study concepts, example questions & explanations for PSAT Math

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Example Questions

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

In the xy -plane, line is given by the equation 2x - 3y = 5. If line passes through the point (a ,1), what is the value of a ?

Possible Answers:
-2
4
-1
5
3
Correct answer: 4
Explanation:

The equation of line l relates x -values and y -values that lie along the line. The question is asking for the x -value of a point on the line whose y -value is 1, so we are looking for the x -value on the line when the y-value is 1. In the equation of the line, plug 1 in for and solve for x:

2x - 3(1) = 5

2x - 3 = 5

2x = 8

x = 4. So the missing x-value on line l is 4.

Example Question #21 : Psat Mathematics

The equation of a line is: 2x + 9y = 71

Which of these points is on that line?

Possible Answers:

(2,7)

(4,-7)

(-4,7)

(-2,7)

(4,7)

Correct answer:

(4,7)

Explanation:

Test the difference combinations out starting with the most repeated number.  In this case, y = 7 appears most often in the answers.  Plug in y=7 and solve for x.  If the answer does not appear on the list, solve for the next most common coordinate.

2(x) + 9(7) = 71

2x + 63 = 71

2x = 8

x = 4

Therefore the answer is (4, 7)

Example Question #1 : Other Lines

Which of the following lines contains the point (8, 9)?

Possible Answers:

\dpi{100} \small 3x-6=2y

\dpi{100} \small 3x+6=y

\dpi{100} \small 8x+9=y

\dpi{100} \small 8x=9y

\dpi{100} \small 3x+6=2y

Correct answer:

\dpi{100} \small 3x-6=2y

Explanation:

In order to find out which of these lines is correct, we simply plug in the values \dpi{100} \small x=8 and \dpi{100} \small y=9 into each equation and see if it balances.

The only one for which this will work is \dpi{100} \small 3x-6=2y

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

\dpi{100} \small 5x+25y = 125

Which point lies on this line?

Possible Answers:

\dpi{100} \small (5,1)

\dpi{100} \small (1,4)

\dpi{100} \small (5,4)

\dpi{100} \small (1,5)

\dpi{100} \small (5,5)

Correct answer:

\dpi{100} \small (5,4)

Explanation:

\dpi{100} \small 5x+25y = 125

Test the coordinates to find the ordered pair that makes the equation of the line true:

\dpi{100} \small (5,4)

\dpi{100} \small 5 (5) + 25 (4) = 25 + 100 = 125

\dpi{100} \small (1,5)

\dpi{100} \small 5(1)+25(5)= 5+125=130

\dpi{100} \small (5,1)

\dpi{100} \small 5(5)+25(1)= 25+25=50

\dpi{100} \small (5,5)

\dpi{100} \small 5(5)+25(5)= 25+125=150

\dpi{100} \small (1,4)

\dpi{100} \small 5(1)+25(4)= 5+100=105

Example Question #32 : Psat Mathematics

Points D and E lie on the same line and have the coordinates and , respectively.  Which of the following points lies on the same line as points D and E?

Possible Answers:

Correct answer:

Explanation:

The first step is to find the equation of the line that the original points, D and E, are on.  You have two points, so you can figure out the slope of the line by plugging the points into the equation 

.  

Therefore, you can get an equation in the line in point-slope form, which is

.  

Plug in the answer options, and you will find that only the point  solves the equation.

Example Question #33 : Psat Mathematics

Which of the following points is on the line given by the equation ?

Possible Answers:

Correct answer:

Explanation:

In order to solve this, try each of the answer choices in the equation:

For example, when we try (3,4), we find:

This does not work. When we try all the choices, we find that only (2,4) works:

Example Question #1 : Points And Distance Formula

One line has four collinear points in order from left to right A, B, C, D.  If AB = 10’, CD was twice as long as AB, and AC = 25’, how long is AD?

Possible Answers:

30'

40'

45'

35'

50'

Correct answer:

45'

Explanation:

AB = 10 ’

BC = AC – AB = 25’ – 10’ = 15’

CD = 2 * AB = 2 * 10’ = 20 ’

AD = AB + BC + CD = 10’ + 15’ + 20’ = 45’

Example Question #1 : How To Find The Length Of A Line With Distance Formula

What is the distance between (1, 4) and (5, 1)?

Possible Answers:

4

9

3

7

5

Correct answer:

5

Explanation:

Let P1 = (1, 4) and P2 = (5, 1)

Substitute these values into the distance formula: 

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The distance formula is an application of the Pythagorean Theorem:  a2 + b2 = c2

Example Question #2 : How To Find The Length Of A Line With Distance Formula

What is the distance of the line drawn between points (–1,–2) and (–9,4)?

Possible Answers:

√5

4

16

10

6

Correct answer:

10

Explanation:

The answer is 10. Use the distance formula between 2 points, or draw a right triangle with legs length 6 and 8 and use the Pythagorean Theorem.

Example Question #3 : How To Find The Length Of A Line With Distance Formula

What is the distance between two points \dpi{100} \small (6,14) and \dpi{100} \small (-6,9)?

Possible Answers:

\dpi{100} \small 5

\dpi{100} \small -12

\dpi{100} \small 13

\dpi{100} \small 17

\dpi{100} \small 10\sqrt{3}

Correct answer:

\dpi{100} \small 13

Explanation:

To find the distance between two points such as these, plot them on a graph.

Then, find the distance between the \dpi{100} \small x units of the points, which is 12, and the distance between the \dpi{100} \small y points, which is 5. The \dpi{100} \small x represents the horizontal leg of a right triangle and the \dpi{100} \small y represents the vertial leg of a right triangle. In this case, we have a 5,12,13 right triangle, but the Pythagorean Theorem can be used as well.

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