How to find the length of the diagonal of a rhombus - ISEE Upper Level (grades 9-12) Quantitative Reasoning
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Question
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
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
Answer
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 
.
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.
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
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.
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