ISEE Middle Level Quantitative : Triangles

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

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Example Questions

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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:

It is impossible to tell from the information given

(a) is greater

(a) and (b) are equal

(b) is greater

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 :

The hypotenuse is 13 feet long. The perimeter is  feet, which is equal to 10 yards.

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) and (b) are equal

(b) is greater

It is impossible to tell from the information given

(a) is greater

Correct answer:

(a) is greater

Explanation:

The length of the second leg of the triangle can be calculated using the Pythagorean Theorem, setting :

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

 centimeters.

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

Example Question #1 : 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

(a) is greater

(b) is greater

It is impossible to tell from the information given

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  inches

(b) 2 feet are equal to 24 inches, so the square has sidelength  inches.

The square has the greater perimeter.

Example Question #632 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

 is an equilateral triangle; .

Rectangle 

Which is the greater quantity?

(a) The perimeter of 

(b) The perimeter of Rectangle 

Possible Answers:

(b) is greater

It is impossible to tell from the information given

(a) is greater

(a) and (b) are equal

Correct answer:

It is impossible to tell from the information given

Explanation:

(a) The perimeter of the equilateral triangle is .

(b)  are of unknown value, but they are equal,  so we will call their common length .

Rectangle  has perimeter

.

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

If , then 

If , then 

 

Example Question #1 : Triangles

 is an isosceles triangle;  is an equilateral triangle

Which is the greater quantity?

(a) The perimeter of

(b) The perimeter of 

Possible Answers:

(b) is greater

It is impossible to tell from the information given

(a) is greater

(a) and (b) are equal

Correct answer:

(a) is greater

Explanation:

(a) As an isosceles triangle, , by definition, has two congruent sides. , so either :

 

in which case the perimeter of  is 

or

in which case the perimeter of  is 

(b)  is an equilateral triangle, so, by definition, all of its sides are congruent; its perimeter is .

Regardless of the length of  has the greater perimeter.

Example Question #6 : Triangles

 and  are right triangles, with right angles , respectively. 

Which is the greater quantity?

(a) 

(b) 

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:

(a)  is the hypotenuse of , so by the Pythagorean Theorem, 

(b)  is a leg of , whose hypotenuse is , so by the Pythagorean Theorem, 

 

Example Question #7 : 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:

It is impossible to tell from the information given

(a) is greater

(b) is greater

(a) and (b) are equal

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 :

The hypotenuse is 5 inches long. The perimeter is therefore  inches, which is equal to one foot.

Example Question #3 : Triangles

 is a right triangle with hypotenuse  10 inches long.

The lengths of  and  , in inches, can both be expressed as integers.

Which is the greater quantity?

(a) 

(b) The perimeter of 

Possible Answers:

It is impossible to tell from the information given

(b) is greater

(a) and (b) are equal

(a) is greater

Correct answer:

(a) and (b) are equal

Explanation:

By the Pythagorean Theorem, 

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:

Add the three side lengths to get the perimeter  inches, which is equal to 2 feet.

Example Question #2 : 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) and (B) are equal

(B) is greater

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

(A) 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  inches. One side has length  inches, which is less than 28 inches, so (B) is greater.

Example Question #5 : Triangles

Right_triangle

Note: Figure NOT drawn to scale.

Refer to the above triangle. An insect walks directly from B to A, then directly from A to C. What percent of the perimeter of the triangle has he walked?

Possible Answers:

It is impossible to answer the question from the information given.

Correct answer:

Explanation:

By the Pythagorean Theorem, the distance from A to C, which we will call , is equal to 

The perimeter of the triangle is . The insect has traveled  units out of 12, which is 

 of the perimeter.

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