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Difficulty: MediumRight Triangle Trigonometry (SOHCAHTOA)

In rhombus ABCDABCD, the diagonals ACAC and BDBD intersect at point EE. If the length of diagonal ACAC is 1616 centimeters and the length of side ABAB is 1010 centimeters, what is the value of sin⁡(∠ABE)\sin(\angle ABE)?

  1. A
    35\frac{3}{5}
  2. 45\frac{4}{5}Answer
  3. C
    34\frac{3}{4}
  4. D
    43\frac{4}{3}
  5. E
    54\frac{5}{4}

Answer

The value of sin⁡(∠ABE)\sin(\angle ABE) is 45\frac{4}{5}.
The correct answer is 45\frac{4}{5}. Because the diagonals of a rhombus are perpendicular bisectors, △ABE\triangle ABE is a right triangle with the right angle at vertex EE. Diagonal ACAC is bisected at EE, making AE=8AE = 8 cm. For ∠ABE\angle ABE, the opposite side is AE=8AE = 8 cm and the hypotenuse is AB=10AB = 10 cm. By SOHCAHTOA, sin⁡(∠ABE)=oppositehypotenuse=810=45\sin(\angle ABE) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{8}{10} = \frac{4}{5}.

Step-by-Step Solution

1
Use the properties of a rhombus to identify right triangles and segment lengths.
The diagonals of a rhombus intersect at right angles (90∘90^\circ) and bisect each other. Therefore, △ABE\triangle ABE is a right triangle with right angle at vertex EE, and leg AE=12AC=12(16)=8AE = \frac{1}{2} AC = \frac{1}{2}(16) = 8 cm.
Rhombus diagonals are perpendicular bisectors.
2
Identify the sides of right triangle ABEABE relative to angle ∠ABE\angle ABE.
The hypotenuse is side AB=10AB = 10 cm, and the side opposite to ∠ABE\angle ABE is leg AE=8AE = 8 cm.
Opposite side is directly across from the angle of interest.
3
Apply the sine ratio formula (SOHCAHTOA).
\sin(\angle ABE) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{AE}{AB} = \frac{8}{10} = \frac{4}{5}
Sine is defined as the ratio of the opposite side length to the hypotenuse length.

Key Concept

Right Triangle Trigonometry (SOHCAHTOA) applied to Rhombus Geometry
Estimated Time:1m 15s
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Topics

Right Triangle Trigonometry (SOHCAHTOA)

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