Points and Distance Formula - PSAT Math
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In the xy -plane, line l is given by the equation 2_x_ - 3_y_ = 5. If line l passes through the point (a ,1), what is the value of a ?
In the xy -plane, line l is given by the equation 2_x_ - 3_y_ = 5. If line l passes through the point (a ,1), what is the value of a ?
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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 y and solve for x:
2_x_ - 3(1) = 5
2_x_ - 3 = 5
2_x_ = 8
x = 4. So the missing x-value on line l is 4.
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 y and solve for x:
2_x_ - 3(1) = 5
2_x_ - 3 = 5
2_x_ = 8
x = 4. So the missing x-value on line l is 4.
The equation of a line is: 2x + 9y = 71
Which of these points is on that line?
The equation of a line is: 2x + 9y = 71
Which of these points is on that line?
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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)
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)
5x+25y = 125
Which point lies on this line?
5x+25y = 125
Which point lies on this line?
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5x+25y = 125
Test the coordinates to find the ordered pair that makes the equation of the line true:
(5,4)
5 (5) + 25 (4) = 25 + 100 = 125
(1,5)
5(1)+25(5)= 5+125=130
(5,1)
5(5)+25(1)= 25+25=50
(5,5)
5(5)+25(5)= 25+125=150
(1,4)
5(1)+25(4)= 5+100=105
5x+25y = 125
Test the coordinates to find the ordered pair that makes the equation of the line true:
(5,4)
5 (5) + 25 (4) = 25 + 100 = 125
(1,5)
5(1)+25(5)= 5+125=130
(5,1)
5(5)+25(1)= 25+25=50
(5,5)
5(5)+25(5)= 25+125=150
(1,4)
5(1)+25(4)= 5+100=105
Which of the following lines contains the point (8, 9)?
Which of the following lines contains the point (8, 9)?
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In order to find out which of these lines is correct, we simply plug in the values x=8 and y=9 into each equation and see if it balances.
The only one for which this will work is 3x-6=2y
In order to find out which of these lines is correct, we simply plug in the values x=8 and y=9 into each equation and see if it balances.
The only one for which this will work is 3x-6=2y
What is the distance between two points (6,14) and (-6,9)?
What is the distance between two points (6,14) and (-6,9)?
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To find the distance between two points such as these, plot them on a graph.
Then, find the distance between the x units of the points, which is 12, and the distance between the y points, which is 5. The x represents the horizontal leg of a right triangle and the 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.
To find the distance between two points such as these, plot them on a graph.
Then, find the distance between the x units of the points, which is 12, and the distance between the y points, which is 5. The x represents the horizontal leg of a right triangle and the 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.
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?
Points D and E lie on the same line and have the coordinates
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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.
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
Which of the following points is on the line given by the equation 
?
Which of the following points is on the line given by the equation
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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:
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:
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?
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?
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AB = 10 ’
BC = AC – AB = 25’ – 10’ = 15’
CD = 2 * AB = 2 * 10’ = 20 ’
AD = AB + BC + CD = 10’ + 15’ + 20’ = 45’
AB = 10 ’
BC = AC – AB = 25’ – 10’ = 15’
CD = 2 * AB = 2 * 10’ = 20 ’
AD = AB + BC + CD = 10’ + 15’ + 20’ = 45’
What is the distance between (1, 4) and (5, 1)?
What is the distance between (1, 4) and (5, 1)?
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Let P1 = (1, 4) and P2 = (5, 1)
Substitute these values into the distance formula:
The distance formula is an application of the Pythagorean Theorem: a2 + b2 = c2
Let P1 = (1, 4) and P2 = (5, 1)
Substitute these values into the distance formula:
The distance formula is an application of the Pythagorean Theorem: a2 + b2 = c2
What is the distance of the line drawn between points (–1,–2) and (–9,4)?
What is the distance of the line drawn between points (–1,–2) and (–9,4)?
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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.
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.
What is the distance between the points 
and 
?
What is the distance between the points
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Plug the points into the distance formula and simplify:
distance2 = (_x_2 – _x_1)2 + (_y_2 – _y_1)2 = (7 – 3)2 + (2 – 12)2 = 42 + 102 = 116
distance = √116 = √(4 * 29) = 2√29
Plug the points into the distance formula and simplify:
distance2 = (_x_2 – _x_1)2 + (_y_2 – _y_1)2 = (7 – 3)2 + (2 – 12)2 = 42 + 102 = 116
distance = √116 = √(4 * 29) = 2√29
Steven draws a line that is 13 units long. If (-4,1) is one endpoint of the line, which of the following might be the other endpoint?
Steven draws a line that is 13 units long. If (-4,1) is one endpoint of the line, which of the following might be the other endpoint?
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The distance formula is $$\sqrt{((x_{2}$$-x_${1})^{2}$ + $(y_{2}$-y_${1})^{2}$)}.
Plug in (-4,1) with each of the answer choices and solve.
Plug in (1,13):
This is therefore the correct answer choice.
The distance formula is $$\sqrt{((x_{2}$$-x_${1})^{2}$ + $(y_{2}$-y_${1})^{2}$)}.
Plug in (-4,1) with each of the answer choices and solve.
Plug in (1,13):
This is therefore the correct answer choice.
What is the distance between (1,3) and (5,6) ?
What is the distance between (1,3) and (5,6) ?
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Let $P_{1}$(1,3) and $P_{2}$(5,6) and use the distance formula:
d = $$\sqrt{(x_{2}$$ - x_${1})^$2$+(y_{2}$ - y_${1})^2$}
Let $P_{1}$(1,3) and $P_{2}$(5,6) and use the distance formula:
d = $$\sqrt{(x_{2}$$ - x_${1})^$2$+(y_{2}$ - y_${1})^2$}
What is the distance between the point 
and the origin?
What is the distance between the point
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The distance between 2 points is found using the distance or Pythagorean Theorem. Because values are squared in the formula, distance can never be a negative value.
The distance between 2 points is found using the distance or Pythagorean Theorem. Because values are squared in the formula, distance can never be a negative value.
What is the distance between (3,4) and (8,16)?
What is the distance between (3,4) and (8,16)?
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The formula for the distance between two points is d = $$\sqrt{(x_{2}$$ - x_${1})^$2$+(y_{2}$ - y_${1})^2$}.
Plug in the points:
d = $$\sqrt{5^2$$+12^2$}$ = 13
The formula for the distance between two points is d = $$\sqrt{(x_{2}$$ - x_${1})^$2$+(y_{2}$ - y_${1})^2$}.
Plug in the points:
d = $$\sqrt{5^2$$+12^2$}$ = 13
Bill gets in his car and drives north for 30 miles at 40 mph. He then turns west and drives 40 mph for 40 minutes. Finally, he goes directly northeast 40 miles in 25 minutes.
Using the total distance traveled as a straight line ("as the crow flies") and the time spent traveling, which of the following is closest to Bill's average speed?
Bill gets in his car and drives north for 30 miles at 40 mph. He then turns west and drives 40 mph for 40 minutes. Finally, he goes directly northeast 40 miles in 25 minutes.
Using the total distance traveled as a straight line ("as the crow flies") and the time spent traveling, which of the following is closest to Bill's average speed?
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Each part of the problem gives you 2 out of the 3 pieces of the rate/time/distance relationship, thus allowing you to find the third (if needed) by using the equation:
The problem is otherwise an application of geometry and the distance formula. We need to find the distance between 2 points, but we need to go step-by-step to find out where the final point is. The first two steps are relatively easy to follow. He travels 30 miles north. Now he turns west and travels for 40 minutes at 40 mph. This is 
of an hour, so we have 
. Thus if we are looking at standard Cartesian coordinates (starting at the origin), we are now at the point 
.
We are now on the last step: 40 miles northeast. We need to decipher this into 
- and 
- coordinate changes. To do this, we think of a triangle. Because we are moving directly northeast, this is a 45 degree angle to the horizontal. We can thus imagine a 45-45-90 triangle with a hypotenuse of 40. Now using the relationships on triangles we have:
So the final step moves us 
up and to the right. Moving this way from the point 
leaves us at:
Using the distance formula for this point from the origin gives us a distance of ~58 miles.
Now for the time. We traveled 30 miles at 40 mph. This means we traveled for 
hours or 45 minutes.
We then traveled 40 mph for 40 minutes. Increasing our total time traveled to 85 minutes.
Finally, we traveled 40 miles in 25 minutes, leaving our total time traveled at 110 minutes. Returning to hours, we have
hours or 1.83 hours.
Our final average speed traveled is then:
The closest answer to this value is 32 mph.
Each part of the problem gives you 2 out of the 3 pieces of the rate/time/distance relationship, thus allowing you to find the third (if needed) by using the equation:
The problem is otherwise an application of geometry and the distance formula. We need to find the distance between 2 points, but we need to go step-by-step to find out where the final point is. The first two steps are relatively easy to follow. He travels 30 miles north. Now he turns west and travels for 40 minutes at 40 mph. This is
We are now on the last step: 40 miles northeast. We need to decipher this into
So the final step moves us
Using the distance formula for this point from the origin gives us a distance of ~58 miles.
Now for the time. We traveled 30 miles at 40 mph. This means we traveled for
We then traveled 40 mph for 40 minutes. Increasing our total time traveled to 85 minutes.
Finally, we traveled 40 miles in 25 minutes, leaving our total time traveled at 110 minutes. Returning to hours, we have
Our final average speed traveled is then:
The closest answer to this value is 32 mph.
In the xy -plane, line l is given by the equation 2_x_ - 3_y_ = 5. If line l passes through the point (a ,1), what is the value of a ?
In the xy -plane, line l is given by the equation 2_x_ - 3_y_ = 5. If line l passes through the point (a ,1), what is the value of a ?
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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 y and solve for x:
2_x_ - 3(1) = 5
2_x_ - 3 = 5
2_x_ = 8
x = 4. So the missing x-value on line l is 4.
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 y and solve for x:
2_x_ - 3(1) = 5
2_x_ - 3 = 5
2_x_ = 8
x = 4. So the missing x-value on line l is 4.
The equation of a line is: 2x + 9y = 71
Which of these points is on that line?
The equation of a line is: 2x + 9y = 71
Which of these points is on that line?
Tap to see back →
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)
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)
5x+25y = 125
Which point lies on this line?
5x+25y = 125
Which point lies on this line?
Tap to see back →
5x+25y = 125
Test the coordinates to find the ordered pair that makes the equation of the line true:
(5,4)
5 (5) + 25 (4) = 25 + 100 = 125
(1,5)
5(1)+25(5)= 5+125=130
(5,1)
5(5)+25(1)= 25+25=50
(5,5)
5(5)+25(5)= 25+125=150
(1,4)
5(1)+25(4)= 5+100=105
5x+25y = 125
Test the coordinates to find the ordered pair that makes the equation of the line true:
(5,4)
5 (5) + 25 (4) = 25 + 100 = 125
(1,5)
5(1)+25(5)= 5+125=130
(5,1)
5(5)+25(1)= 25+25=50
(5,5)
5(5)+25(5)= 25+125=150
(1,4)
5(1)+25(4)= 5+100=105
Which of the following lines contains the point (8, 9)?
Which of the following lines contains the point (8, 9)?
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In order to find out which of these lines is correct, we simply plug in the values x=8 and y=9 into each equation and see if it balances.
The only one for which this will work is 3x-6=2y
In order to find out which of these lines is correct, we simply plug in the values x=8 and y=9 into each equation and see if it balances.
The only one for which this will work is 3x-6=2y