Geometry - ISEE Middle Level Quantitative Reasoning

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Question

The area of a square is ?

Answer

The area of a square is the side length squared not the side length times .

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Question

If you are given one side length of a square, you can find the area with that information.

Answer

To find the area of a square, you multiple . But with a square all the sides are equal so the equation really is or the side length squared. Since you are given the side length, you can find the area.

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Question

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

Answer

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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Question

Which is the greater quantity?

(A) The slope of the line

(B) The slope of the line

Answer

Rewrite each in the slope-intercept form, ; will be the slope.

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

The slope of the line of is

The slope of the line of is also

The slopes are equal.

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Question

Which is the greater quantity?

(A) The slope of the line

(B) The slope of the line

Answer

Rewrite each in the slope-intercept form, ; will be the slope.

The slope of this line is .

The slope of this line is .

Since , (A) is greater.

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Question

Each side of a square is units long. Which is the greater quantity?

(A) The area of the square

(B)

Answer

The area of a square is the square of its side length:

Using the side length from the question:

$A=\left ( 5t\right ) ^{2} = 5 ^{2} \cdot t ^{2} = 25t ^{2}$

However, it is impossible to tell with certainty which of $25t ^{2}$ and is greater.

For example, if ,

$25t ^{2} = 25 \cdot 2^{2} = 25 \cdot4 = 100$

and

$25t = 25 \cdot 2 = 50$

so $25t ^{2} > 25t$ if .

But if $t = \frac{1}{5}$ ,

$25t ^{2}=25 \cdot \left ( \frac{1}{5} \right )^{2} = 25 \cdot \frac{1}{25} = 1$

and

$25t =25 \cdot \frac{1}{5} = 5$

so $25t ^{2} < 25t$ if $t = \frac{1}{5}$ .

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Question

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

(a) The slope of the line on the coordinate plane through the points and .

(b) The slope of the line on the coordinate plane through the points and .

Answer

The slope of a line through the points and 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 and 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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Question

A line passes through the points with coordinates and , where . Which expression is equal to the slope of the line?

Answer

The slope of a line through the points and , 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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Question

Choose the best answer from the four choices given.

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

Answer

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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Question

Choose the best answer from the four choices given.

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

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

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

Answer

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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Question

Choose the best answer from the four choices given.

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

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

Answer

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

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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Question

The ordered pair is in which quadrant?

Answer

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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Question

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

Answer

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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Question

The lines of the equations

and

intersect at a point .

Which is the greater quantity?

(a)

(b)

Answer

If and , we can substitute in the second equation as follows:

$4x \div 4 = 8 \div 4$

Substitute:

$y= 3 \cdot 2$

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Question

What is the perimeter of a square with area 196 square inches?

Answer

A square with area 196 square inches has sidelength $\sqrt{196} = 14$ inches, and therefore has perimeter $4 \cdot 14 = 56$ inches

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Question

If a square has sides measuring $\small \small \frac{1}{8}$ , what is the perimeter of the square, in simplest form?

Answer

To find the perimeter of a square, you must add together all the sides. In this case, we are adding $\small \frac{1}{8}$ four times.

$\small \frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}=\frac{4}{8}$

Since all of the denominators are the same, there is no need to find a commond denominator, so we add together the numerators. This gives us $\small \frac{4}{8}$ .

Since both the numerator and denomator are divisible by four, we must simplify this fraction.

$\small \frac{4/4}{8/4}= \frac{1}{2}$

The perimeter of the square is $\small \small \frac{1}{2}$ .

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Question

The sum of the lengths of three sides of a square is one foot. Give the perimeter of the square in inches.

Answer

A square has four sides of the same length.

One foot is equal to twelve inches; since the sum of the lengths of three of the congruent sides is twelve inches, each side measures

$12 \div 3 = 4$ inches.

The perimeter is

$4 \times 4 = 16$ inches.

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Question

A square has perimeter one yard. Which is the greater quantity?

(A) The length of one side of the square

(B) 8 inches

Answer

One yard is equal to 36 inches. A square has four sides of equal length, so one side of the square has length

$36 \div 4 = 9$ inches.

Since , (A) is greater.

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Question

A square has perimeter five meters. Which is the greater quantity?

(A) 1,250 millimeters

(B) The length of one side of the square

Answer

One meter is equal to 1,000 millimeters, so the square has perimeter

$5 \times 1,000 = 5,000$ millimeters.

A square has four sides of equal length, so one side of the square has length

$5,000 \div 4 = 1,250$ millimeters.

The quantities are equal.

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Question

A square has perimeter one meter. Which is the greater quantity?

(A) 250 centimeters

(B) The length of one side of the square

Answer

One meter is equal to 100 centimeters.

A square has four sides of equal length, so we will need to divide by 4 to find the length of one side.

$P\div4=s$

$100cm \div 4 = 25cm$

, so (A) is greater

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