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SSAT Upper Level Math : Geometry

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Example Questions

Example Question #53 : Area Of Polygons

Parallelogram

Figure NOT drawn to scale

The above figure shows Rhombus ABCD ; and are midpoints of their respective sides. Rhombus ABCD has area 900.

Give the area of Rectangle AXCY .

Possible Answers:

450

300

675

225

Correct answer:

450

Explanation:

A rhombus, by definition, has four sides of equal length. Therefore, AB= CD , and, by the Multiplication Property, $\frac{1}{2 }AB=\frac{1}{2 } CD$ . Also, since and are the midpoints of their respective sides, $AX = XB = \frac{1}{2 }AB$ and $\frac{1}{2 } CD = CY = YD$ . Combining these statements, and letting n = AX :

$n = AX = XB = \frac{1}{2 }AB=\frac{1}{2 } CD = CY = YD$

Also, both $\overline{AY}$ and $\overline{XC}$ are altitudes of the rhombus; they are congruent, and we will call their common length (height).

The figure, with the lengths, is below.

Rhombus

The area of the entire Rhombus ABCD is the product of its height and the length of a base , so

$A_{1} = h \cdot 2n = 2hn =900$ .

Rectangle AXCY has as its length and width and , so its area is their product , Since

2 hn =900 ,

From the Division Property, it follows that

$2 hn \div 2 = 900 \div 2$ ,

and

hn = 450 .

This makes 450 the area of Rectangle AXCY .

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Example Question #2 : Area Of A Parallelogram

A parallelogram has the base length of and the altitude of 2t-1 . Give the area of the parallelogram.

Possible Answers:

t^2+t

t^2

2t^2+t

2t^2

2t^2-t

Correct answer:

2t^2-t

Explanation:

The area of a parallelogram is given by:

Area=ba

Where is the base length and is the corresponding altitude. So we can write:

Area=t(2t-1)=2t^2-t

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Example Question #2 : Area Of A Parallelogram

A parallelogram has a base length of which is 3 times longer than its corresponding altitude. The area of the parallelogram is 12 square inches. Give the .

Possible Answers:

$6\sqrt{2}$

$6\sqrt{3}$

Correct answer:

Explanation:

Base length is so the corresponding altitude is $\frac{t}{3}$ .

The area of a parallelogram is given by:

Area=ba

Where:

is the length of any base
is the corresponding altitude

So we can write:

$12=t\times \frac{t}{3}\Rightarrow t\times t=12\times 3\Rightarrow t^2=36\Rightarrow t=6$

$t\times t=12\times 3$

$t^{2}=36$

t=6

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Example Question #4 : Area Of A Parallelogram

The length of the shorter diagonal of a rhombus is 40% that of the longer diagonal. The area of the rhombus is . Give the length of the longer diagonal in terms of .

Possible Answers:

$\frac{1}{5}\sqrt{K}$

$K\sqrt{\frac{5}{2}}$

$\sqrt{\frac{5K}{2}}$

$\sqrt{5K}$

$K\sqrt{\frac{2}{5}}$

Correct answer:

$\sqrt{5K}$

Explanation:

Let be the length of the longer diagonal. Then the shorter diagonal has length 40% of this. Since 40% is equal to $\frac{40}{100 } = \frac{40 \div 20 }{100 \div 20 } = \frac{2}{5}$ , 40% of is equal to $\frac{2}{5}D$ .

The area of a rhombus is half the product of the lengths of its diagonals, so we can set up, and solve for , in the equation:

$\frac{1}{2} \cdot \frac{2}{5}D \cdot D = K$

$\frac{1}{5}D^{2}= K$

$5\cdot \frac{1}{5}D^{2}=5\cdot K$

$D^{2}=5K$

$D =\sqrt{5K}$

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Example Question #4 : Area Of A Parallelogram

The length of the shorter diagonal of a rhombus is two-thirds that of the longer diagonal. The area of the rhombus is square yards. Give the length of the longer diagonal, in inches, in terms of .

Possible Answers:

$\sqrt{ 3Q}$

$3\sqrt{ 3Q}$

$36\sqrt{ 3Q }$

$4\sqrt{ 3Q}$

$12\sqrt{ 3Q}$

Correct answer:

$36\sqrt{ 3Q }$

Explanation:

Let be the length of the longer diagonal in yards. Then the shorter diagonal has length two-thirds of this, or $\frac{2}{3}D$ .

The area of a rhombus is half the product of the lengths of its diagonals, so we can set up the following equation and solve for :

$\frac{1}{2} \cdot \frac{2}{3}D \cdot D = Q$

$\frac{1}{3}D ^{2} = Q$

$3\cdot \frac{1}{3}D ^{2} = 3 \cdot Q$

$D ^{2} = 3Q$

$D = \sqrt{3Q}$

To convert yards to inches, multiply by 36:

$\sqrt{ 3Q } \times 36 = 36\sqrt{ 3Q }$

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Example Question #2 : Area Of A Parallelogram

The longer diagonal of a rhombus is 20% longer than the shorter diagonal; the rhombus has area . Give the length of the shorter diagonal in terms of .

Possible Answers:

$\sqrt{ \frac{3N}{5}}$

$\sqrt{ \frac{5N}{3}}$

$\sqrt{ \frac{N}{5}}$

$\sqrt{15N}$

$\sqrt{5N}$

Correct answer:

$\sqrt{ \frac{5N}{3}}$

Explanation:

Let be the length of the shorter diagonal. If the longer diagonal is 20% longer, then it measures 120% of the length of the shorter diagonal; this is

$\frac{120}{100} = \frac{120 \div 20}{100 \div 20} = \frac{6}{5}$

of , or $\frac{6}{5}D$ .

The area of a rhombus is half the product of the lengths of its diagonals, so we can set up an equation and solve for :

$\frac{1}{2} \cdot \frac{6}{5}D \cdot D = N$

$\frac{3}{5}D^{2} = N$

$\frac{5}{3} \cdot \frac{3}{5}D^{2} = \frac{5}{3} \cdot N$

$D^{2} = \frac{5N}{3}$

$D =\sqrt{ \frac{5N}{3}}$

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

Which of the following shapes is NOT a quadrilateral?

Possible Answers:

Kite

Triangle

Square

Rhombus

Rectangle

Correct answer:

Triangle

Explanation:

A quadrilateral is any two-dimensional shape with sides. The only shape listed that does not have sides is a triangle.

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Example Question #1 : Understand Categories And Subcategories Of Two Dimensional Figures: Ccss.Math.Content.5.G.B.3

What is the main difference between a square and a rectangle?

Possible Answers:

Their side lengths

The number of sides they each have

The sum of their angles

Their color

Their angle measurments

Correct answer:

Their side lengths

Explanation:

The only difference between a rectangle and a square is their side lengths. A square has to have equal side lengths, but the opposite side lengths of a rectangle only have to be equal.

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Example Question #2 : Understand Categories And Subcategories Of Two Dimensional Figures: Ccss.Math.Content.5.G.B.3

What is the main difference between a triangle and a rectangle?

Possible Answers:

The volume

The length of the sides

The area

The color

The number of sides

Correct answer:

The number of sides

Explanation:

Out of the choices given, the only characteristic used to describe shapes is the number of sides. A triangle has sides and a rectangle has sides.

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Example Question #4 : How To Find The Area Of A Parallelogram

Which two shapes have to have right angles?

Possible Answers:

Rectangle and Parallelogram

Square and Rectangle

Rectangle and Rhombus

Square and Parallelogram

Square and Rhombus

Correct answer:

Square and Rectangle

Explanation:

By definition, the only two quadrilaterals that have to have right angles, are the square and the rectangle.

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SSAT Upper Level Math : Geometry

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

All SSAT Upper Level Math Resources

Example Questions

Example Question #53 : Area Of Polygons

Example Question #2 : Area Of A Parallelogram

Example Question #2 : Area Of A Parallelogram

Example Question #4 : Area Of A Parallelogram

Example Question #4 : Area Of A Parallelogram

Example Question #2 : Area Of A Parallelogram

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

Example Question #1 : Understand Categories And Subcategories Of Two Dimensional Figures: Ccss.Math.Content.5.G.B.3

Example Question #2 : Understand Categories And Subcategories Of Two Dimensional Figures: Ccss.Math.Content.5.G.B.3

Example Question #4 : How To Find The Area Of A Parallelogram

All SSAT Upper Level Math Resources

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