PSAT Math : Geometry

Study concepts, example questions & explanations for PSAT Math

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Example Questions

Example Question #1 : How To Find The Equation Of A Perpendicular Line

What line is perpendicular to 2x+5y=9 through (4,12)?

Possible Answers:

2x-5y=12

-5x+2y=4

2x+5y=11

-3x-2y=5

5x+2y=7

Correct answer:

-5x+2y=4

Explanation:

We need to find the slope of the given equation by converting it to the slope intercept form:  y=-\frac{2}{5}x+\frac{9}{5}

The slope is -\frac{2}{5} and the perpendicular slope would be the opposite reciprocal, or \frac{5}{2}

The new equation is of the form y=\frac{5}{2}x+b and we can use the point (4,12) to calculate b=2. The next step is to convert y=\frac{5}{2}x+2 into the standard form of -5x+2y=4.

Example Question #10 : How To Find The Equation Of A Perpendicular Line

What line is perpendicular to y=-\frac{1}{3}x+2 and passes through (1,7)?

Possible Answers:

x+3y=2

-3x-y=5

x+y=7

-3x+y=4

2x-3y=-3

Correct answer:

-3x+y=4

Explanation:

Perpendicular slopes are opposite reciprocals.  The original slope is -\frac{1}{3} so the new perdendicular slope is 3.

We plug the point (1,7) and the slope m=3 into the point-slope form of the equation:

y-y_{1}=m(x-x_{1})

to get y=3x+4 or in standard form -3x+y=4.

Example Question #11 : How To Find The Equation Of A Perpendicular Line

Solve the system of equations for the point of intersection.

 

 

Possible Answers:

Correct answer:

Explanation:

First one needs to use one of the two equations to substitute one of the unknowns.

From the second equation we can derive that y = x – 3.

Then we substitute what we got into the first equation which gives us: x + x – 3 = 15.

Next we solve for x, so 2x = 18 and x = 9.

x – y = 3, so = 6.

These two lines will intersect at the point (9,6).

                 

 

 

Example Question #11 : How To Find The Equation Of A Perpendicular Line

Line A is perpendicular to y=2x+1 and passes the point (0,1). Find the -intercept of line A.

Possible Answers:

2.5

1.5

0.5

 

1

2

Correct answer:

2

Explanation:

We are given an equation of a line and told that line A is perpendicular to it.  The slope of the given line is 2.  Therefore, the slope of line A must be -0.5, since perpendicular lines have slopes that are negative reciprocals of each other.

The equation for line A will therefore take the form , where b is the y-intercept.

Since we are told that it crosses (0,1), we can plug in the point and solve for c:

Then the equation becomes y=-0.5x+1.

To find the x-intercept, plug in 0 for y and solve for x:

x=2

Example Question #71 : Psat Mathematics

What line is perpendicular to  through ?

Possible Answers:

Correct answer:

Explanation:

The slope of the given line is , and the slope of the perpendicular line is its negative reciprocal, .  We take the new slope and the given point  and plug them into the slope-intercept form of a line, .

 

Thus, the perpendicular line has the equation , or in standard form, .

Example Question #11 : How To Find The Equation Of A Perpendicular Line

In the xy-plane, the equation of the line n is –x+8y=17. If the line m is perpendicular to line n, what is a possible equation of line m?

 

Possible Answers:

y= -1/8x + 5

y= 8x-17

y= -8x + 5

x= -8y + (17/8)

Correct answer:

y= -8x + 5

Explanation:

We start by add x to the other side of the equation to get the y by itself, giving us 8y =17 + x. We then divide both sides by 8, giving us y= 17/8 + 1/8x. Since we are looking for the equation of a perpendicular line, we know the slope (the coefficient in front of x) will be the opposite reciprocal of the slope of our line, giving us y= -8x + 5 as the answer.

 

 

Example Question #1 : How To Find The Equation Of A Perpendicular Line

What line is perpendicular to x + 3y = 6 and travels through point (1,5)?

Possible Answers:

y = –1/3x – 4

y = 6x – 3

y = 2/3x + 6

y = 3x + 2

y = 2x + 1

Correct answer:

y = 3x + 2

Explanation:

Convert the equation to slope intercept form to get y = –1/3x + 2.  The old slope is –1/3 and the new slope is 3.  Perpendicular slopes must be opposite reciprocals of each other:  m1 * m2 = –1

With the new slope, use the slope intercept form and the point to calculate the intercept: y = mx + b or 5 = 3(1) + b, so b = 2

So y = 3x + 2

Example Question #1 : How To Find The Equation Of A Perpendicular Line

What line is perpendicular to and passes through ?

Possible Answers:

Correct answer:

Explanation:

Convert the given equation to slope-intercept form.

The slope of this line is . The slope of the line perpendicular to this one will have a slope equal to the negative reciprocal.

The perpendicular slope is .

Plug the new slope and the given point into the slope-intercept form to find the y-intercept.

So the equation of the perpendicular line is .

Example Question #1 : How To Find The Equation Of A Perpendicular Line

What is the equation of a line that runs perpendicular to the line 2x + = 5 and passes through the point (2,7)?

Possible Answers:

x/2 + y = 5

2x – y = 6

x/2 + y = 6

2x + y = 7

x/2 – y = 6

Correct answer:

x/2 + y = 6

Explanation:

First, put the equation of the line given into slope-intercept form by solving for y. You get y = 2x +5, so the slope is –2. Perpendicular lines have opposite-reciprocal slopes, so the slope of the line we want to find is 1/2. Plugging in the point given into the equation y = 1/2x + b and solving for b, we get b = 6. Thus, the equation of the line is y = ½x + 6. Rearranged, it is –x/2 + y = 6.

Example Question #1 : How To Find The Equation Of A Perpendicular Line

Line m passes through the points (1, 4) and (5, 2). If line p is perpendicular to m, then which of the following could represent the equation for p?

Possible Answers:

2x + y = 3

3x + 2y = 4

x  y = 3

2x  y = 3

4x  3y = 4

Correct answer:

2x  y = 3

Explanation:

The slope of m is equal to   y2-y1/x2-x1  =  2-4/5-1 -1/2                                  

Since line p is perpendicular to line m, this means that the products of the slopes of p and m must be 1:

 

(slope of p) * (-1/2) = -1

               

Slope of p = 2

So we must choose the equation that has a slope of 2. If we rewrite the equations in point-slope form (y = mx + b), we see that the equation 2x  y = 3 could be written as y = 2x – 3. This means that the slope of the line 2x – y =3 would be 2, so it could be the equation of line p. The answer is 2x – y = 3.

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