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ISEE Upper Level Quantitative : Plane Geometry

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Example Questions

Example Question #31 : Geometry

$\Delta ABC$ is an equilateral triangle. Points D,E,F are the midpoints of $\overline{AB},\overline{BC},\overline{AC}$ , respectively. $\Delta DEF$ is constructed.

Which is the greater quantity?

(a) The area of $\Delta ABC$

(b) Twice the area of $\Delta DEF$

Possible Answers:

(b) is greater.

(a) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

Correct answer:

(a) is greater.

Explanation:

If segments are constructed in which the endpoints form the midpoints of the sides of a triangle, then four triangles, congruent to each other and similar to the larger triangle, are formed. Therefore, one of these triangles - specifically, $\Delta DEF$ - would have one-fourth the area of $\Delta ABC$ . This means $\Delta ABC$ has more than twice the area of $\Delta DEF$ .

Note that the fact that the triangle is equilateral is irrelevant.

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Example Question #1 : How To Find The Length Of The Side Of An Equilateral Triangle

Which of the following could be the three sidelengths of an equilateral triangle?

Possible Answers:

$\frac{2}{3} \textrm{ yd, } 24 \textrm{ in, }2\textrm{ ft}$

$1\frac{1}{3} \textrm{ yd, } 48 \textrm{ in, }5\textrm{ ft}$

$2 \textrm{ yd, } 84 \textrm{ in, }7\textrm{ ft}$

$1\frac{1}{2} \textrm{ yd, } 50 \textrm{ in, }4\textrm{ ft}$

$1\frac{2}{3} \textrm{ yd, } 48 \textrm{ in, }4\textrm{ ft}$

Correct answer:

$\frac{2}{3} \textrm{ yd, } 24 \textrm{ in, }2\textrm{ ft}$

Explanation:

By definition, an equilateral triangle has three sides of equal length. We can identify the equilateral triangle by converting the given sidelengths to the same units and comparing them.

We can eliminate the following by showing that at least two sidelengths differ.

$2 \textrm{ yd, } 84 \textrm{ in, }7\textrm{ ft}$

2 yards = $2 \times 3 = 6$ feet.

Two sides have lengths 6 feet and 7 feet, so we can eliminate this choice.

$1\frac{1}{2} \textrm{ yd, } 50 \textrm{ in, }4\textrm{ ft}$

4 feet = $4 \times 12 = 48$ inches

Two sides have lengths 48 inches and 50 inches, so we can eliminate this choice.

$1\frac{1}{3} \textrm{ yd, } 48 \textrm{ in, }5\textrm{ ft}$

5 feet = $5 \times 12 = 60$ inches

Two sides have lengths 48 inches and 60 inches, so we can eliminate this choice.

$1\frac{2}{3} \textrm{ yd, } 48 \textrm{ in, }4\textrm{ ft}$

$1\frac{2}{3}$ yards = $1\frac{2}{3} \times 3 = 5$ feet

Two sides have lengths 4 feet and 5 feet, so we can eliminate this choice.

$\frac{2}{3} \textrm{ yd, } 24 \textrm{ in, }2\textrm{ ft}$

$\frac{2}{3}$ yards = $\frac{2}{3} \times3 = 2$ feet = $2 \times 12 = 24$ inches

All three sides have the same length, making this the triangle equilateral. This choice is correct.

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Example Question #1 : Right Triangles

What is the hypotenuse of a right triangle with sides 5 and 8?

Possible Answers:

√89

5√4

Correct answer:

√89

Explanation:

Because this is a right triangle, we can use the Pythagorean Theorem which says a² + b² = c², or the squares of the two sides of a right triangle must equal the square of the hypotenuse. Here we have a = 5 and b = 8.

a² + b² = c²

5² + 8² = c²

25 + 64 = c²

89 = c²

c = √89

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Example Question #31 : Plane Geometry

Which is the greater quantity?

(a) The hypotenuse of a $45^{\circ}-45^{\circ}-90^{\circ}$ right triangle with a leg of length 20

(b) The hypotenuse of a right triangle with legs of length 19 and 21

Possible Answers:

(a) and (b) are equal

(b) is greater

(a) is greater

It is impossible to tell from the information given

Correct answer:

(b) is greater

Explanation:

The hypotenuses of the triangles measure as follows:

(a) $c = \sqrt {20^{2} + 20^{2} } = \sqrt {400 + 400 } = \sqrt {800}$

(b) $c = \sqrt {19^{2} + 21^{2} } = \sqrt {361 + 441} = \sqrt {802}$

800 < 802 , so $\sqrt {800} < \sqrt {802}$ , making (b) the greater quantity

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Example Question #3 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Which is the greater quantity?

(a) The hypotenuse of a right triangle with legs and .

(b) The hypotenuse of a right triangle with legs and .

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) is greater.

Explanation:

The hypotenuses of the triangles measure as follows:

(a) $c = \sqrt {10^{2} + 15^{2} } = \sqrt {100 + 225 } = \sqrt {325}$

(b) $c = \sqrt {12^{2} + 13^{2} } = \sqrt {144 + 169 } = \sqrt {313}$

325 > 313 , so $\sqrt {325} > \sqrt {313}$ , making (a) the greater quantity.

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Example Question #4 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

A right triangle has a leg $4 \frac{1}{2}$ feet long and a hypotenuse $7 \frac{1}{2}$ feet long. Which is the greater quantity?

(a) The length of the second leg of the triangle

(b) 60 inches

Possible Answers:

(a) is greater.

(b) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

Correct answer:

(a) is greater.

Explanation:

The length of the second leg can be calculated using the Pythagorean Theorem. Set $c = 7 \frac{1}{2} = \frac{15}{2} , a =4 \frac{1}{2} = \frac{9}{2}$ :

$b ^{2} = c ^{2}-a ^{2}$

$b ^{2} = \left ( \frac{15}{2} \right ) ^{2}- \left ( \frac{9}{2} \right )^{2}\textup{}$

$b ^{2} = \frac{225}{4} -\frac{81}{4}$

$b ^{2} = \frac{144}{4}$

$b ^{2} = 36$

$b = \sqrt{36} = 6$

The second leg therefore measures $6 \times 12 = 72$ inches.

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Example Question #5 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

What is the hypotenuse of a right triangle with sides 9 inches and 12 inches?

Possible Answers:

25in

10.5in

27in

225in

15 in

Correct answer:

15 in

Explanation:

Since we're dealing with right triangles, we can use the Pythagorean Theorem ( a^2+b^2=c^2 ). In this formula, a and b are the sides, while c is the hypotenuse. The hypotenuse of a right triangle is the longest side and the side that is opposite the right angle. Now, we can plug into our formula, which looks like this: 9^2+12^2=c^2. We simplify and get 225=c^2 . At this point, isolate c. This means taking the square root of both sides so that your answer is 15in.

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Example Question #6 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Right_triangle

The perimeter of a regular pentagon is 75% of that of the triangle in the above diagram. Which is the greater quantity?

(A) The length of one side of the pentagon

(B) One and one-half feet

Possible Answers:

(A) is greater

(B) is greater

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

(A) and (B) are equal

Correct answer:

(B) is greater

Explanation:

By the Pythagorean Theorem, the hypotenuse of the right triangle is

$\sqrt{14^{2}+48^{2}} = \sqrt{196+2,304} = \sqrt{2,500} = 50$ inches, making its perimeter

14 + 48 + 50 =112 inches.

The pentagon in question has sides of length 75% of 112, or

$112 \times 0.75 = 84$ .

Since a pentagon has five sides of equal length, each side will have measure

$84 \div 5 = 16 \frac{4}{5}$ inches.

One and a half feet are equivalent to $12 \times 1 \frac{1}{2} = 18$ inches, so (B) is the greater quantity.

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Example Question #7 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Right_triangle

The track at Gauss High School is unusual in that it is shaped like a right triangle, as shown above.

Cary decides to get some exercise by running from point A to point B, then running half of the distance from point B to point C.

Which is the greater quantity?

(A) The distance Cary runs

(B) One-fourth of a mile

Possible Answers:

(A) and (B) are equal

(B) is greater

(A) is greater

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

Correct answer:

(B) is greater

Explanation:

By the Pythagorean Theorem, the distance from B to C is

$\sqrt{600^{2} + 800^{2} }$

$= \sqrt{360,000 + 640,000 }$

$= \sqrt{1,000,000} = 1,000$ feet

Cary runs

$800 + \frac{1}{2} \times1,000 = 800 + 500 = 1,300$ feet

Since 5,280 feet make a mile, one-fourth of a mile is equal to

$5,280 \div 4 = 1,320$ feet.

(B) is greater

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Example Question #8 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Right_triangle

Give the length of the hypotenuse of the above right triangle in terms of .

Possible Answers:

$2k ^{2} +8k$

$\sqrt{2k ^{2} +8}$

$\sqrt{2k ^{2} +8k}$

$2 \sqrt{2k}$

$2k ^{2} +8$

Correct answer:

$\sqrt{2k ^{2} +8}$

Explanation:

If we let be the length of the hypotenuse, then by the Pythagorean theorem,

$c = \sqrt{(k+2)^{2}+(k-2)^{2}}$

$c = \sqrt{(k ^{2}+4k+4) +(k ^{2}-4k+4)}$

$c = \sqrt{2k ^{2} +8}$

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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : Plane Geometry

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

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #31 : Geometry

Example Question #1 : How To Find The Length Of The Side Of An Equilateral Triangle

Example Question #1 : Right Triangles

Example Question #31 : Plane Geometry

Example Question #3 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #4 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #5 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #6 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #7 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #8 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

All ISEE Upper Level Quantitative Resources

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