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Difficulty: MediumProperties of Quadrilaterals

In rhombus JKLMJKLM, diagonals JLJL and KMKM intersect at point PP. If JP=2x+3JP = 2x + 3, PL=4x−5PL = 4x - 5, and m∠JKL=60∘m\angle JKL = 60^\circ, what is the length of side JKJK?

  1. A
    88
  2. B
    1111
  3. 2222Answer
  4. D
    11311\sqrt{3}
  5. E
    4444

Answer

The length of side JKJK is 2222.
Since the diagonals of a rhombus bisect each other, segment JPJP equals segment PLPL. Equating 2x+3=4x−52x + 3 = 4x - 5 yields x=4x = 4, which gives JP=11JP = 11. The diagonals of a rhombus are perpendicular and bisect the vertex angles, creating right triangle △JPK\triangle JPK with m∠JPK=90∘m\angle JPK = 90^\circ and m∠JKP=30∘m\angle JKP = 30^\circ. In a 30∘30^\circ-60∘60^\circ-90∘90^\circ right triangle, the hypotenuse is twice the side opposite the 30∘30^\circ angle. Because JP=11JP = 11 is opposite the 30∘30^\circ angle, side JK=2×11=22JK = 2 \times 11 = 22. Alternatively, since △JKL\triangle JKL is an isosceles triangle with a 60∘60^\circ vertex angle, it is equilateral, making JK=JL=22JK = JL = 22.

Step-by-Step Solution

1
Use the diagonal bisecting property of a rhombus to set up an algebraic equation.
2x+3=4x−5  ⟹  2x=8  ⟹  x=42x + 3 = 4x - 5 \implies 2x = 8 \implies x = 4
The diagonals of a rhombus bisect each other, so JP=PLJP = PL.
2
Calculate the length of the half-diagonal segment JPJP.
JP=2(4)+3=11JP = 2(4) + 3 = 11
Substitute x=4x = 4 into the expression for JPJP.
3
Determine the angle measures in right triangle △JPK\triangle JPK.
m∠JPK=90∘m\angle JPK = 90^\circ and m∠JKP=12(60∘)=30∘m\angle JKP = \frac{1}{2}(60^\circ) = 30^\circ
The diagonals of a rhombus are perpendicular to each other and bisect the vertex angles.
4
Apply the 30∘30^\circ-60∘60^\circ-90∘90^\circ right triangle ratio or sine function to find hypotenuse JKJK.
\sin(30^\circ) = \frac{JP}{JK} \implies \frac{1}{2} = \frac{11}{JK} \implies JK = 22$
In right triangle △JPK\triangle JPK, JPJP is opposite the 30∘30^\circ angle, so the hypotenuse JKJK is twice the length of JPJP.

Key Concept

Properties of Rhombus Diagonals and Special Right Triangles
Estimated Time:1m 30s
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Topics

Properties of Quadrilaterals

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