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ISEE Upper Level Math : Geometry

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Example Questions

Example Question #291 : Isee Upper Level (Grades 9 12) Mathematics Achievement

Find the area of a rhombus with one diagonal having a length of 12in and the other having a length two times the first diagonal.

Possible Answers:

$100\text{in}^2$

$288\text{in}^2$

$144\text{in}^2$

$121\text{in}^2$

$48\text{in}^2$

Correct answer:

$144\text{in}^2$

Explanation:

To find the area of a rhombus, we will use the following formula:

$A = \frac{pq}{2}$

where p and q are the lengths of the diagonals of the rhombus.

Now, we know one diagonal has a length of 12in. We also know the other diagonal is two times the first diagonal. Therefore, the second diagonal is 24in.

Knowing this, we can substitute into the formula. We get

$A = \frac{12\text{in} \cdot 24\text{in}}{2}$

$A = \frac{288\text{in}^2}{2}$

$A = 144\text{in}^2$

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Example Question #1 : How To Find The Length Of The Side Of A Rhombus

A rhombus has a perimeter of $36\ in$ . Give the side length of the rhombus.

Possible Answers:

$8\ in$

$6\ in$

$9\ in$

$10\ in$

$12\ in$

Correct answer:

$9\ in$

Explanation:

Like any polygon, the perimeter of the rhombus is the total distance around the outside, which can be found by adding together the length of each side. In the case of a rhombus, all four sides have the same length, i.e.

Perimeter=4a ,

where is the length of each side.

$Perimeter=4a\Rightarrow 36=4a\Rightarrow a=9\ in$

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Example Question #1 : Rhombuses

A rhombus has the area of $9\sqrt{2}\ cm^2$ . One of the interior angles is $45^{\circ}$ . Give the length of each side of the rhombus.

Possible Answers:

$3\ cm$

$2\sqrt{2}\ cm$

$3\sqrt{3}\ cm$

$2\ cm$

$3\sqrt{2}\ cm$

Correct answer:

$3\sqrt{2}\ cm$

Explanation:

The area of a rhombus can be determined by the following formula:

$Area=s^2sin\alpha$ ,

where is the length of any side and $\alpha$ is any interior angle.

$Area=s^2sin\alpha\Rightarrow 9\sqrt{2}=s^2\times sin45^{\circ}\Rightarrow 9\sqrt{2}=s^2 \times \frac{\sqrt{2}}{2}$

$\Rightarrow s^2=18\Rightarrow s=\sqrt{18}=\sqrt{9\times 2}=3\sqrt{2}\ cm$

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Example Question #1 : How To Find The Perimeter Of A Rhombus

One side of a rhombus measures 2 yards. Which of the following is equal to the perimeter?

Possible Answers:

$\frac{1}{200} \textrm{ mi}$

$12 \textrm{ yd}$

All of the other choices give the correct perimeter.

$18 \textrm{ ft}$

$288 \textrm{ in }$

Correct answer:

$288 \textrm{ in }$

Explanation:

A rhombus has four sides of equal measure, so the perimeter of a rhombus with sidelength 2 yards is $2 \times 4 = 8$ yards.

$8 \textrm{ yd} = 8 \times 3 = 24 \textrm{ ft} = 24 \times 12 = 288 \textrm{ in}$

Also,

$8 \textrm{ yd} = 24 \textrm{ ft} =\frac{ 24}{5,280} = \frac{1}{220} \textrm{ mi}$

The only choice that agrees with any of these measures is 288 inches, the correct choice.

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Example Question #2 : How To Find The Perimeter Of A Rhombus

One side of a rhombus measures y + 12 . Give its perimeter in terms of .

Possible Answers:

$y^{2}+ 12y + 144$

3y + 36

$y^{2}+ 24y + 144$

4y + 48

2y + 24

Correct answer:

4y + 48

Explanation:

A rhombus has four sides of equal measure, so the perimeter of a rhombus with sidelength y + 12 is 4 (y + 12) = 4y + 48 .

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Example Question #1 : How To Find An Angle In A Rhombus

Consider the rhombus below. Solve for .

Problem_10

Possible Answers:

x=145

x=325

x=35

x=65

Correct answer:

x=145

Explanation:

The total sum of the interior angles of a quadrilateral is 360 degrees. In this problem, we are only considering half of the interior angles:

$\frac{360}{2}=180$

35+x=180

x=145

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Example Question #1 : How To Find An Angle In A Rhombus

Rhombus

Note: Figure NOT drawn to scale.

The above depicts a rhombus and one of its diagonals. What is ?

Possible Answers:

x = 29

x = 48

x = 58

x = 42

x = 32

Correct answer:

x = 29

Explanation:

The diagonals of a rhombus bisect the angles.

The angle bisected must be supplementary to the $122^{\circ }$ angle since they are consecutive angles of a parallelogram; therefore, that angle has measure $\left (180 - 122 \right ) ^{\circ } = 58^{\circ }$ , and is half that, or $29^{\circ }$ .

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Example Question #1 : How To Find The Area Of A Rectangle

A rectangle on the coordinate plane has its vertices at the points (3,4), (-6,4), (-6,-2), (3,-2) .

What percent of the rectangle is located in Quadrant I?

Possible Answers:

$33 \frac{1}{3} \%$

$22 \frac{2}{9} \%$

$11 \frac{1}{9} \%$

$44 \frac{4}{9} \%$

$25 \%$

Correct answer:

$22 \frac{2}{9} \%$

Explanation:

The total area of the rectangle is

$\left [ 3- (-6) \right ] \times\left [ 4- (-2) \right ] = 9 \times 6 = 54$ .

The area of the portion of the rectangle in Quadrant I is

$\left ( 3- 0 \right ) \times\left ( 4- 0 \right) = 3 \times 4 = 12$ .

Therefore, the portion of the rectangle in Quadrant I is

$\frac{12}{54} = \frac{2}{9} = \frac{2}{9} \times 100 \% = 22 \frac{2}{9} \%$ .

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Example Question #1 : Rectangles

A rectangle and a square have the same perimeter. The area of the square is 225 square centimeters; the length of the rectangle is centimeters. Give the width of the rectangle in centimeters.

Possible Answers:

$10 \frac{5}{7}\textrm{ cm}$

$18\textrm{ cm}$

$9\textrm{ cm}$

$21 \frac{3}{7}\textrm{ cm}$

$6\textrm{ cm}$

Correct answer:

$9\textrm{ cm}$

Explanation:

The sidelength of a square with area 225 square centimeters is $\sqrt{225} = 15$ centimeters; its perimeter, as well as that of the rectangle, is therefore $15 \times 4 = 60$ centimeters.

Using the formula for the perimeter of a rectangle, substitute L = 21 and solve for as follows:

2L + 2W = P

$2 \cdot 21 + 2W = 60$

42 + 2W = 60

42 -42 + 2W = 60-42

2W = 18

$2W \div 2 = 18\div 2$

W = 9

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Example Question #1 : Rectangles

In a rectangle, the width is while the length is . If 3x=9 , what is the area of the rectangle?

Possible Answers:

Correct answer:

Explanation:

In a rectangle in which the width is x while the length is 4x, the first step is to solve for x. If 3x=9 , the value of x can be found by dividing each side of this equation by 3.

Doing so gives us the information that x is equal to 3.

Thus, the area is equal to:

$Area=x\cdot4x$

$Area=3\cdot4\cdot3$

Area=36

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All ISEE Upper Level Math Resources

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ISEE Upper Level Math : Geometry

Study concepts, example questions & explanations for ISEE Upper Level Math

All ISEE Upper Level Math Resources

Example Questions

Example Question #291 : Isee Upper Level (Grades 9 12) Mathematics Achievement

Example Question #1 : How To Find The Length Of The Side Of A Rhombus

Example Question #1 : Rhombuses

Example Question #1 : How To Find The Perimeter Of A Rhombus

Example Question #2 : How To Find The Perimeter Of A Rhombus

Example Question #1 : How To Find An Angle In A Rhombus

Example Question #1 : How To Find An Angle In A Rhombus

Example Question #1 : How To Find The Area Of A Rectangle

Example Question #1 : Rectangles

Example Question #1 : Rectangles

All ISEE Upper Level Math Resources

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