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Difficulty: MediumPythagorean Theorem and Special Right Triangles

A rhombus-shaped garden plot ABCDABCD has side lengths of 1515 feet each. The length of the shorter diagonal, ACAC, is 1818 feet. A gardener places a straight divider line along the longer diagonal, BDBD. What is the length, in feet, of the divider line along diagonal BDBD?

Answer: 24 feet

Answer

The length of the divider line along diagonal BDBD is 24 feet.
The diagonals of a rhombus are perpendicular bisectors of each other. The point of intersection EE creates right triangle AEBAEB, where the hypotenuse is rhombus side AB=15AB = 15 feet and one leg is AE=182=9AE = \frac{18}{2} = 9 feet. Applying the Pythagorean Theorem yields 92+BE2=1529^2 + BE^2 = 15^2, which simplifies to 81+BE2=22581 + BE^2 = 225, giving BE=12BE = 12 feet. Doubling BEBE gives the complete length of diagonal BD=24BD = 24 feet.

Step-by-Step Solution

1
Identify geometric properties of a rhombus regarding its diagonals.
The diagonals of rhombus ABCDABCD are perpendicular to each other and bisect each other at intersection point EE.
In any rhombus, the diagonals act as perpendicular bisectors, forming four right triangles.
2
Calculate the leg length AEAE in right triangle AEBAEB.
AE=182=9AE = \frac{18}{2} = 9 feet.
Point EE is the midpoint of diagonal ACAC.
3
Use the Pythagorean Theorem to calculate leg length BEBE.
92+BE2=152  ⟹  81+BE2=225  ⟹  BE2=144  ⟹  BE=129^2 + BE^2 = 15^2 \implies 81 + BE^2 = 225 \implies BE^2 = 144 \implies BE = 12 feet.
In right triangle AEBAEB, side AB=15AB = 15 is the hypotenuse, and AE=9AE = 9 is one leg.
4
Find the total length of diagonal BDBD.
BD=2×BE=2×12=24BD = 2 \times BE = 2 \times 12 = 24 feet.
Point EE bisects diagonal BDBD, so BDBD is twice the length of BEBE.

Key Concept

Using the Pythagorean Theorem on right triangles formed by the perpendicular bisecting diagonals of a rhombus.
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Topics

Pythagorean Theorem and Special Right Triangles

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