How to find the length of the diagonal of a rhombus

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ISEE Upper Level Quantitative Reasoning › How to find the length of the diagonal of a rhombus

Questions 1 - 1

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

(B) is greater

(A) is greater

(A) and (B) are equal

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

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 $\frac{n}{2}$ .

Rhombus_2

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

$\frac{n}{2} = 8$

$\frac{n}{2} \cdot 2 = 8 \cdot 2$

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