ISEE Upper Level Quantitative : Rhombuses

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

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Example Questions

Example Question #1 : Rhombuses

In Rhombus , and . Which is the greater quantity?

(A)

(B)

Possible Answers:

(a) and (b) are equal

(b) is the greater quantity

It cannot be determined which of (a) and (b) is greater

(a) is the greater quantity

Correct answer:

(a) is the greater quantity

Explanation:

The four sides of a rhombus, by defintion, have equal length, so

Since and are positive, .

Example Question #61 : Quadrilaterals

A rhombus has diagonals of length two and one-half feet and six feet. Which is the greater quantity?

(A) The perimeter of the rhombus

(B) Four yards

Possible Answers:

(B) is greater

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

(A) is greater

 (A) and (B) are equal

Correct answer:

(A) is greater

Explanation:

It will be easier to look at these measurements as inches for the time being:

 and , so these are the lengths of the diagonals in inches.

The diagonals of a rhombus are each other's perpendicular bisector, so, as can be seen in the diagram below, one side of a rhombus and one half of each diagonal form a right triangle. If we let  be the length of one side of the rhombus, then this is the hypotenuse of that right triangle; its legs are one-half the lengths of the diagonals, or 15 and 36 inches.

Rhombus_2

By the Pythagorean Theorem, 

 

Each side of the rhombus measures 39 inches, and its perimeter is 

 inches.

Four yards is equal to  inches, so (A) is greater.

Example Question #62 : Quadrilaterals

A rhombus has sidelength ten inches; one of its diagonals is one foot long. Which is the greater quantity?

(a) The length of the other diagonal

(B) One and one-half feet

Possible Answers:

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

(A) is greater

(B) is greater

(A) and (B) are equal

Correct answer:

(B) is greater

Explanation:

The diagonals of a rhombus are each other's perpendicular bisector, so, as can be seen in the diagram below, one side of a rhombus and one half of each diagonal form a right triangle. If the other diagonal has length , then the right triangle has hypotenuse 10 inches and legs one-half of one foot and  - that is, six inches and .

Rhombus_2

This triangle fits the well-known Pythagorean triple of 6-8-10, so

 

The other diagonal measures 16 inches. One and one-half feet is equal to 18 inches, making (B) greater.

 

Example Question #4 : Rhombuses

Rhombus  has two diagonals that intersect at point . Which is the greater quantity?

(a) 

(b) 

Possible Answers:

(b) is greater

(a) and (b) are equal

It is impossible to tell from the information given

(a) is greater

Correct answer:

(a) and (b) are equal

Explanation:

The diagonals of a rhombus always intersect at right angles, so . The measures of the interior angles of the rhombus are irrelevant.

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