Geometry - ISEE Middle Level Quantitative Reasoning

Card 0 of 1110

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

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

Note: Figure NOT drawn to scale

The above figure shows Square .

Which is the greater quantity?

(a) The area of Trapezoid

(b) The area of Trapezoid

Answer

The easiest way to answer the question is to locate on $\overline{SQ}$ such that :

Trapezoids and have the same height, which is . Their bases, by construction, have the same lengths - and . Therefore, Trapezoids and have the same area.

Since , it follows that , and . It follows that Trapezoid is greater in area than Trapezoids and , and Trapezoid is less in area.

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Question

A triangle has base 80 inches and area 4,200 square inches. What is its height?

Answer

Use the area formula for a triangle, setting :

$A = \frac{1}{2} bh$

$4200 = \frac{1}{2}\cdot 80 \cdot h = 40h$

$h = 4200 \div 40 = 105$ inches

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Question

The sum of the lengths of the legs of an isosceles right triangle is one meter. What is its area in square centimeters?

Answer

The legs of an isosceles right triangle have equal length, so, if the sum of their lengths is one meter, which is equal to 100 centimeters, each leg measures half of this, or

$100 \div 2 = 50$ centimeters.

The area of a triangle is half the product of its height and base; for a right triangle, the legs serve as height and base, so the area of the triangle is

$\frac{1}{2} \times 50 \times 50 = 1,250$ square centimeters.

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Question

Figure NOT drawn to scale

Square has area 1,600. ; . Which of the following is the greater quantity?

(a) The area of $\bigtriangleup AZX$

(b) The area of $\bigtriangleup BZY$

Answer

Square has area 1,600, so the length of each side is $\sqrt{1,600 }= 40$ .

Since ,

Therefore, .

$\bigtriangleup AZX$ has as its area $\frac{1}{2} \cdot AX \cdot AZ$ ; $\bigtriangleup BZY$ has as its area $\frac{1}{2} \cdot BX \cdot BZ$ .

Since and , it follows that

$BX \cdot BZ > AX \cdot AZ$

and

$\frac{1}{2} \cdot BX \cdot BZ >\frac{1}{2} \cdot AX \cdot AZ$

$\bigtriangleup BZY$ has greater area than $\bigtriangleup AZX$ .

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Question

The above figure depicts Square . , , and are the midpoints of $\overline{AD}$ , $\overline{BC}$ , and $\overline{AB}$ , respectively.

$\bigtriangleup AZX$ has area . What is the area of Square ?

Answer

Since , , and are the midpoints of $\overline{AD}$ , $\overline{BC}$ , and $\overline{AB}$ , if we call the length of each side of the square, then

$AX = AZ = \frac{1}{2}s$

The area of $\bigtriangleup AZX$ is half the product of the lengths of its legs:

$\frac{1}{2} \cdot AX \cdot AZ$

$= \frac{1}{2} \cdot \frac{1}{2}s \cdot \frac{1}{2}s$

$= \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2}\cdot s \cdot s$

$= \frac{1 \cdot 1 \cdot 1}{2\cdot 2 \cdot 2}\cdot s ^{2}$

$= \frac{1}{8} s ^{2}$

The area of the square is the square of the length of a side, which is $s^{2}$ . This is eight times the area of $\bigtriangleup AZX$ , so the correct choice is

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Question

Which of the following is the greater quantity?

(a) The area of the above triangle

(b) 800

Answer

The area of a right triangle is half the product of the lengths of its legs, which here are 25 and 60. So

$A = \frac{1}{2} \cdot 25 \cdot 60 = 750$

which is less than 800.

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Question

The above figure gives the lengths of the three sides of the triangle in feet. Give its area in square inches.

Answer

The area of a right triangle is half the product of the lengths of its legs, which here are feet and feet.

Multiply each length by 12 to convert to inches - the lengths become and . The area in square inches is therefore

$\frac{1}{2} (12A )(12 B) = \frac{1}{2} \cdot 12 \cdot 12 \cdot A \cdot B = 72AB$ square inches.

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Question

Refer to the above figure. Which is the greater quantity?

(a) The perimeter of the triangle

(b) 3 feet

Answer

The perimeter of the triangle - the sum of the lengths of its sides - is

inches.

3 feet are equivalent to $3 \times 12 = 36$ inches, so this is the greater quantity.

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Question

Figure NOT drawn to scale.

In the above diagram, Square has area 400. Which is the greater quantity?

(a) The area of $\bigtriangleup SQE$

(b) The area of $\bigtriangleup UAR$

Answer

Square has area 400, so its common sidelength is the square root of 400, or 20. Therefore,

.

The area of a right triangle is half the product of the lengths of its legs.

$\bigtriangleup SQE$ has legs $\overline{SQ}$ and $\overline{SE}$ , so its area is

$\frac{1}{2} \cdot SQ \cdot SE = \frac{1}{2} \cdot 9 \cdot 20 = 90$ .

$\bigtriangleup UAR$ has legs $\overline{UA}$ and $\overline{AR}$ , so its area is

$\frac{1}{2} \cdot AR \cdot UA = \frac{1}{2} \cdot 11 \cdot 20 = 110$ .

$\bigtriangleup UAR$ has the greater area.

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