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ISEE Middle Level Quantitative : Geometry

Study concepts, example questions & explanations for ISEE Middle Level Quantitative

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All ISEE Middle Level Quantitative Resources

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Example Questions

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

Square c

Note: Figure NOT drawn to scale

The above figure shows Square SQUA .

SX = UY + 1

Which is the greater quantity?

(a) The area of Trapezoid SXYA

(b) The area of Trapezoid XQUY

Possible Answers:

(b) is the greater quantity

(a) is the greater quantity

(a) and (b) are equal

It is impossible to determine which is greater from the information given

Correct answer:

(a) is the greater quantity

Explanation:

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

Square c

Trapezoids SZYA and ZQUY have the same height, which is SA = QU . Their bases, by construction, have the same lengths - AY = ZQ and SZ = YU . Therefore, Trapezoids SZYA and ZQUY have the same area.

Since SX = UY + 1 , it follows that SX = SZ+ 1 , and SX > SZ . It follows that Trapezoid SXYA is greater in area than Trapezoids SZYA and ZQUY , and Trapezoid XQUY is less in area.

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Example Question #1 : Triangles

Which is the greater quantity?

(a) The perimeter of a right triangle with legs of length 5 feet and 12 feet

(b) 8 yards

Possible Answers:

(a) is greater

(a) and (b) are equal

(b) is greater

It is impossible to tell from the information given

Correct answer:

(a) is greater

Explanation:

The length of the hypotenuse of a triangle with legs 5 feet and 12 feet is calculated using the Pythagorean Theorem, setting a = 5, b=12 :

$c = \sqrt{a^{2}+b^{2}} = \sqrt{5^{2}+12^{2}} = \sqrt{25+144} = \sqrt{169} =13$

The hypotenuse is 13 feet long. The perimeter is 5 + 12 + 13 = 30 feet, which is equal to 10 yards.

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Example Question #2 : Triangles

Which is the greater quantity?

(a) The perimeter of a right triangle with hypotenuse of length 25 centimeters and one leg of length 7 centimeters

(b) One-half of a meter

Possible Answers:

(a) is greater

(a) and (b) are equal

(b) is greater

It is impossible to tell from the information given

Correct answer:

(a) is greater

Explanation:

The length of the second leg of the triangle can be calculated using the Pythagorean Theorem, setting a = 7,c =25 :

$b = \sqrt{c^{2}-a^{2}} = \sqrt{25^{2}-7^{2}} = \sqrt{625-49}= \sqrt{576} = 24$

The second leg has length 24 centimeters, so the perimeter of the triangle is

7 + 24 + 25 = 56 centimeters.

One-half of a meter is one-half of 100 centimeters, or 50 centimeters, so (a) is greater.

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Example Question #3 : Triangles

Which is the greater quantity?

(a) The perimeter of an equilateral triangle with sidelength 30 inches

(b) The perimeter of a square with sidelength 2 feet

Possible Answers:

(a) and (b) are equal

It is impossible to tell from the information given

(a) is greater

(b) is greater

Correct answer:

(b) is greater

Explanation:

Each figure has sides that are congruent, so in each case, multiply the sidelength by the number of sides.

(a) The triangle has perimeter $30 \times 3 = 90$ inches

(b) 2 feet are equal to 24 inches, so the square has sidelength $24 \times 4 = 96$ inches.

The square has the greater perimeter.

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Example Question #1 : Triangles

$\Delta ABC$ is an equilateral triangle; AB = 25 .

Rectangle RECT ; RE = 35

Which is the greater quantity?

(a) The perimeter of $\Delta ABC$

(b) The perimeter of Rectangle RECT

Possible Answers:

It is impossible to tell from the information given

(a) and (b) are equal

(a) is greater

(b) is greater

Correct answer:

It is impossible to tell from the information given

Explanation:

(a) The perimeter of the equilateral triangle is $25 \times 3 = 75$ .

(b) CT = RE = 35 ; EC, RT are of unknown value, but they are equal, so we will call their common length .

Rectangle RECT has perimeter

35 + 35 + N + N = 70 + 2N .

Without knowing , it cannot be determined with certainty which figure has the longer perimeter. For example:

If N = 1 , then $70 + 2N = 70 + 2 \cdot 1 = 70 + 2 = 72 < 75$

If N = 3 , then $70 + 2N = 70 + 2 \cdot 3 = 70 + 6 = 76>75$

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Example Question #2 : Triangles

$\Delta ABC$ is an isosceles triangle; $\Delta DEF$ is an equilateral triangle

AB = 30,BC = 50, DE= 35

Which is the greater quantity?

(a) The perimeter of $\Delta ABC$

(b) The perimeter of $\Delta DEF$

Possible Answers:

(a) and (b) are equal

It is impossible to tell from the information given

(a) is greater

(b) is greater

Correct answer:

(a) is greater

Explanation:

(a) As an isosceles triangle, $\Delta ABC$ , by definition, has two congruent sides. $AB \neq BC$ , so either :

AC = AB = 30

in which case the perimeter of $\Delta ABC$ is

AB + AC + BC = 30 + 30 + 50 = 110

AC =BC = 50

in which case the perimeter of $\Delta ABC$ is

AB + AC + BC = 30 + 50 + 50 = 130

(b) $\Delta DEF$ is an equilateral triangle, so, by definition, all of its sides are congruent; its perimeter is $35 \times 3 = 105$ .

Regardless of the length of $\overline{AC}$ , $\Delta ABC$ has the greater perimeter.

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Example Question #4 : Triangles

$\Delta ABC$ and $\Delta DEF$ are right triangles, with right angles $\angle B , \angle E$ , respectively.

AB = 6,BC = 8, DE = 24, DF = 26

Which is the greater quantity?

(a)

(b)

Possible Answers:

(a) is greater

It is impossible to tell from the information given

(b) is greater

(a) and (b) are equal

Correct answer:

(a) and (b) are equal

Explanation:

(a) $\overline{AC}$ is the hypotenuse of $\Delta ABC$ , so by the Pythagorean Theorem,

$\left (AC \right )^{2} =\left ( AB \right ) ^{2} + \left ( B C\right )^{2}$

$\left (AC \right )^{2} =6^{2} + 8^{2}$

$\left (AC \right )^{2} =36 + 64 = 100$

$AC = \sqrt{100} = 10$

(b) $\overline{EF}$ is a leg of $\Delta DEF$ , whose hypotenuse is $\overline{DF}$ , so by the Pythagorean Theorem,

$\left (EF \right )^{2} =\left ( DF \right ) ^{2} - \left ( DE\right )^{2}$

$\left (EF \right )^{2} =26^{2} - 24^{2}$

$\left (EF \right )^{2} =676 -576 = 100$

$EF= \sqrt{100} = 10$

AC = EF

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Example Question #4 : Triangles

Which is the greater quantity?

(a) The perimeter of a right triangle with legs of length 3 inches and 4 inches

(b) One foot

Possible Answers:

(a) and (b) are equal

It is impossible to tell from the information given

(b) is greater

(a) is greater

Correct answer:

(a) and (b) are equal

Explanation:

The length of the hypotenuse of a triangle with legs 3 inches and 4 inches long is calculated using the Pythagorean Theorem, setting a = 3, b=4 :

$c = \sqrt{a^{2}+b^{2}} = \sqrt{3^{2}+4^{2}} = \sqrt{9+16} = \sqrt{25} =5$

The hypotenuse is 5 inches long. The perimeter is therefore 3 + 4 + 5 = 12 inches, which is equal to one foot.

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Example Question #235 : Plane Geometry

$\Delta ABC$ is a right triangle with hypotenuse $\overline{AC}$ 10 inches long.

The lengths of $\overline{AB}$ and $\overline{BC}$ , in inches, can both be expressed as integers.

Which is the greater quantity?

(a) $2 \textrm{ feet}$

(b) The perimeter of $\Delta ABC$

Possible Answers:

(b) is greater

It is impossible to tell from the information given

(a) and (b) are equal

(a) is greater

Correct answer:

(a) and (b) are equal

Explanation:

By the Pythagorean Theorem,

$(AB)^{2}+ (BC)^{2}= (AC)^{2}$

$(AB)^{2}+ (BC)^{2}= 10^{2}$

$(AB)^{2}+ (BC)^{2}= 100$

By trial and error, it can be determined that the only two positive integers that can replace and to make this equation true are 6 and 8, in either order:

$6^{2}+ 8^{2}= 100$

36+64 = 100

Add the three side lengths to get the perimeter 6 + 8 + 10 = 24 inches, which is equal to 2 feet.

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Example Question #5 : Triangles

An equilateral triangle has perimeter two yards. Which is the greater quantity?

(A) The length of one side of the triangle

(B) 28 inches

Possible Answers:

(A) is greater

(A) and (B) are equal

It is impossible to determine which is greater from the information given

(B) is greater

Correct answer:

(B) is greater

Explanation:

An equilateral triangle has three sides of equal length; the perimeter of this triangle is two yards, which is equal to $2 \times 36 = 72$ inches. One side has length $72 \div 3 = 24$ inches, which is less than 28 inches, so (B) is greater.

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ISEE Middle Level Quantitative : Geometry

Study concepts, example questions & explanations for ISEE Middle Level Quantitative

All ISEE Middle Level Quantitative Resources

Example Questions

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

Example Question #1 : Triangles

Example Question #2 : Triangles

Example Question #3 : Triangles

Example Question #1 : Triangles

Example Question #2 : Triangles

Example Question #4 : Triangles

Example Question #4 : Triangles

Example Question #235 : Plane Geometry

Example Question #5 : Triangles

All ISEE Middle Level Quantitative Resources

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