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ISEE Middle Level Quantitative : Geometry

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Example Questions

Example Question #1 : Geometry

Give the equation of the line through point (1,4) that has slope $-\frac{1}{5}$ .

Possible Answers:

$y = \frac{1}{5}x - \frac{19}{5}$

$y = -\frac{1}{5}x + 4$

$y = -\frac{1}{5}x + \frac{21}{5}$

y = -5x + 9

y = -5x + 4

Correct answer:

$y = -\frac{1}{5}x + \frac{21}{5}$

Explanation:

Use the point-slope formula with $m = -\frac{1}{5}, x_{1}= 1, y_{1} = 4$

$y -y _{1} = m (x -x _{1} )$

$y -4 = -\frac{1}{5} (x -1 )$

$y -\frac{20}{5} = -\frac{1}{5} x + \frac{1}{5}$

$y = -\frac{1}{5} x + \frac{21}{5}$

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Example Question #2 : Geometry

Which is the greater quantity?

(A) The slope of the line 2x+y = 15

(B) The slope of the line 3x+y = 15

Possible Answers:

It is impossible to determine which is greater from the information given

(B) is greater

(A) and (B) are equal

(A) is greater

Correct answer:

(A) is greater

Explanation:

Rewrite each in the slope-intercept form, y = mx + b ; will be the slope.

2x+y = 15

2x+y -2x = 15-2x

y=15-2x

y = -2x+ 15

The slope of this line is m = -2 .

3x+y = 15

3x+y -3x = 15-3x

y=15-3x

y = -3x+ 15

The slope of this line is m = -3 .

Since -2 > -3 , (A) is greater.

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Example Question #2 : Coordinate Geometry

Which is the greater quantity?

(A) The slope of the line 4x - 2y = 10

(B) The slope of the line y - 2x = 7

Possible Answers:

(A) is greater

(A) and (B) are equal

It is impossible to determine which is greater from the information given

(B) is greater

Correct answer:

(A) and (B) are equal

Explanation:

Rewrite each in the slope-intercept form, y = mx + b ; will be the slope.

4x - 2y = 10

4x - 2y -4x = 10-4x

- 2y = -4x+ 10

$- 2y \div (-2) =\left ( -4x+ 10 \right )\div (-2)$

y = 2x -5

The slope of the line of 4x - 2y = 10 is m = 2

y - 2x = 7

y - 2x + 2x = 7+ 2x

y = 2x + 7

The slope of the line of y - 2x = 7 is also m = 2

The slopes are equal.

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Example Question #2 : Geometry

and are positive integers, and A > B . Which is the greater quantity?

(a) The slope of the line on the coordinate plane through the points (A+ 1, B+ 1 ) and (A-1, B- 1 ) .

(b) The slope of the line on the coordinate plane through the points (B- 1, A- 1 ) and (B+1, A+ 1 ) .

Possible Answers:

(b) is the greater quantity

It is impossible to determine which is greater from the information given

(a) and (b) are equal

(a) is the greater quantity

Correct answer:

(a) and (b) are equal

Explanation:

The slope of a line through the points (A+ 1, B+ 1 ) and (A-1, B- 1 ) can be found by setting

$x_{1} = A-1,y_{1} = B - 1, x_{2} = A+1, y_{2} = B+ 1$

in the slope formula:

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

$= \frac{ (B+ 1)- (B- 1)}{ (A+ 1)- (A- 1)}$

$= \frac{ B-B+ 1+1}{ A-A + 1 + 1}$

$= \frac{ 2}{ 2}$

The slope of a line through the points (B- 1, A- 1 ) and (B+1, A+ 1 ) can be found similarly:

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

$= \frac{(A+ 1)- (A- 1) }{ (B+ 1)- (B- 1) }$

$= \frac{A-A + 1+1}{B-B + 1 + 1}$

$= \frac{ 2}{ 2}$

The lines have the same slope.

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Example Question #3 : Coordinate Geometry

A line passes through the points with coordinates (A, B) and (-A, B ) , where A > B >0 . Which expression is equal to the slope of the line?

Possible Answers:

$\frac{B}{A}$

$\frac{A}{B}$

Undefined

Correct answer:

Explanation:

The slope of a line through the points (A, B) and (-A, B ) , can be found by setting

$x_{1} = -A,y_{1} =B, x_{2} = A, y_{2} = B$ :

in the slope formula:

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

$m = \frac{B-B}{A - (-A)} = \frac{0}{A+A} = \frac{0}{2A} = 0$

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Example Question #1 : How To Find The Points On A Coordinate Plane

Choose the best answer from the four choices given.

The point (15, 6) is on which of the following lines?

Possible Answers:

$y=\frac{2}{3}x-4$

$y=-\frac{2}{3}x+4$

$y=\frac{1}{2}x-7$

$y=\frac{-1}{2}x+7$

Correct answer:

$y=\frac{2}{3}x-4$

Explanation:

For this problem, simply plug in the values for the point (15,6) into the different equations (15 for the -value and 6 for the -value) to see which one fits.

$(6)=\frac{1}{2}(15)-7$ (NO)

$(6)=\frac{2}{3}(15)-4$ (YES!)

$(6)=\frac{-1}{2}(15)+7$ (NO)

$(6)=\frac{-2}{3}(15)+4$ (NO)

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Example Question #2 : How To Find The Points On A Coordinate Plane

Choose the best answer from the four choices given.

What is the point of intersection for the following two lines?

$y=\frac{3}{4}x-11$

$y=\frac{-5}{6}x+8$

Possible Answers:

$\left ( 12,-2 \right )$

$\left ( 8,-5 \right )$

$\left ( 6,3 \right )$

$\left ( 0,-11 \right )$

Correct answer:

$\left ( 12,-2 \right )$

Explanation:

At the intersection point of the two lines the - and - values for each equation will be the same. Thus, we can set the two equations as equal to each other:

$\frac{3}{4}x-11=\frac{-5}{6}x+8$

$\frac{3}{4}x=\frac{-5}{6}x+19$

$\frac{9}{12}x=\frac{-10}{12}x+19$

$\frac{19}{12}x=19$

$(\frac{12}{19})(\frac{19}{12}x)=19 (\frac{12}{19})$

$\large x=12$

$y=\frac{3}{4}(12)-11= -2$

point of intersection $=\left ( 12,-2 \right )$

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Example Question #3 : How To Find The Points On A Coordinate Plane

Choose the best answer from the four choices given.

What is the -intercept of the line represented by the equation

$y=\frac{7}{9}x-3\ ?$

Possible Answers:

$3\frac{6}{7}$

$-3\frac{6}{7}$

Correct answer:

$3\frac{6}{7}$

Explanation:

In the formula y=mx+b , the y-intercept is represented by (because if you set to zero, you are left with y=b ).

Thus, to find the -intercept, set the value to zero and solve for .

$0=\frac{7}{9}x-3$

$\frac{-7}{9}x=-3$

$(\frac{-9}{7})\frac{-7}{9}x=-3 (\frac{-9}{7})$

$x=\frac{27}{7}=3\frac{6}{7}$

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Example Question #5 : Coordinate Geometry

The ordered pair (-1,4) is in which quadrant?

Possible Answers:

Quadrant I

Quadrant III

Quadrant V

Quadrant II

Quadrant IV

Correct answer:

Quadrant II

Explanation:

There are four quadrants in the coordinate plane. Quadrant I is the top right, and they are numbered counter-clockwise. Since the x-coordinate is , you go to the left one unit (starting from the origin). Since the y-coordinate is , you go upwards four units. Therefore, you are in Quadrant II.

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Example Question #1 : Coordinate Geometry

If angles s and r add up to 180 degrees, which of the following best describes them?

Possible Answers:

Complementary

Acute

Obtuse

Supplementary.

Correct answer:

Supplementary.

Explanation:

Two angles that are supplementary add up to 180 degrees. They cannot both be acute, nor can they both be obtuse. Therefore, "Supplementary" is the correct answer.

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ISEE Middle Level Quantitative : Geometry

Study concepts, example questions & explanations for ISEE Middle Level Quantitative

All ISEE Middle Level Quantitative Resources

Example Questions

Example Question #1 : Geometry

Example Question #2 : Geometry

Example Question #2 : Coordinate Geometry

Example Question #2 : Geometry

Example Question #3 : Coordinate Geometry

Example Question #1 : How To Find The Points On A Coordinate Plane

Example Question #2 : How To Find The Points On A Coordinate Plane

Example Question #3 : How To Find The Points On A Coordinate Plane

Example Question #5 : Coordinate Geometry

Example Question #1 : Coordinate Geometry

All ISEE Middle Level Quantitative Resources

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