Geometry - ISEE Middle Level Quantitative Reasoning

Card 0 of 2265

Question

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

Answer

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

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

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

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

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

Compare your answer with the correct one above

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.

Compare your answer with the correct one above

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.

Compare your answer with the correct one above

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 $t = \frac{1}{5}$ .

Compare your answer with the correct one above

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.

Compare your answer with the correct one above

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:

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

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

Compare your answer with the correct one above

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)

Compare your answer with the correct one above

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

Compare your answer with the correct one above

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 .

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

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

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

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

Compare your answer with the correct one above

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.

Compare your answer with the correct one above

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.

Compare your answer with the correct one above

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:

$y= 3 \cdot 2$

Compare your answer with the correct one above

Question

What is the area of the figure below?

Answer

To find the area of the figure above, we need to slip the figure into two rectangles.

Using our area formula, $A=l\times w$ , we can solve for the area of both of our rectangles

$A=3\times 3$ $A=2\times 1$

To find our final answer, we need to add the areas together.

Compare your answer with the correct one above

Question

What is the area of the figure below?

Answer

To find the area of the figure above, we need to slip the figure into two rectangles.

Using our area formula, $A=l\times w$ , we can solve for the area of both of our rectangles

$A=6\times 8$ $A=7\times 5$

To find our final answer, we need to add the areas together.

Compare your answer with the correct one above

Question

The sum of the lengths of three sides of a square is one yard. Give its area in square inches.

Answer

A square has four sides of the same length.

One yard is equal to 36 inches, so each side of the square has length

$36 \div 3 = 12$ inches.

Its area is the square of the sidelength, or

$12^{2} = 12 \times 12 = 144$ square inches.

Compare your answer with the correct one above

Question

What is the area of the figure below?

Answer

To find the area of the figure above, we need to split the figure into two rectangles.

Using our area formula, $A=l\times w$ , we can solve for the area of both of our rectangles

$A=7\times 5$ $A=6\times 4$

To find our final answer, we need to add the areas together.

Compare your answer with the correct one above

Question

What is the area of the figure below?

Answer

To find the area of the figure above, we need to slip the figure into two rectangles.

Using our area formula, $A=l\times w$ , we can solve for the area of both of our rectangles

$A=6\times 2$ $A=5\times 2$

To find our final answer, we need to add the areas together.

Compare your answer with the correct one above

Question

What is the area of the figure below?

Answer

To find the area of the figure above, we need to slip the figure into two rectangles.

Using our area formula, $A=l\times w$ , we can solve for the area of both of our rectangles

$A=7\times 2$ $A=6\times 3$

To find our final answer, we need to add the areas together.

Compare your answer with the correct one above

Question

What is the area of the figure below?

Answer

To find the area of the figure above, we need to slip the figure into two rectangles.

Using our area formula, $A=l\times w$ , we can solve for the area of both of our rectangles

$A=9\times 8$ $A=8\times 3$

To find our final answer, we need to add the areas together.

Compare your answer with the correct one above

Question

What is the area of the figure below?

Answer

To find the area of the figure above, we need to slip the figure into two rectangles.

Using our area formula, $A=l\times w$ , we can solve for the area of both of our rectangles

$A=4\times 6$ $A=4\times 2$

To find our final answer, we need to add the areas together.

Compare your answer with the correct one above

Tap the card to reveal the answer