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Zorluk: OrtaProperties of Quadrilaterals

In rhombus ABCDABCD, the measure of interior angle ∠DAB\angle DAB is 120∘120^\circ, and the length of diagonal ACAC is 1212 inches. What is the perimeter, in inches, of rhombus ABCDABCD?

  1. A
    24
  2. 48Cevap
  3. C
    48348\sqrt{3}
  4. D
    72
  5. E
    96

Cevap

The perimeter of rhombus ABCDABCD is 48 inches.
Since consecutive angles in a rhombus are supplementary, ∠ABC=180∘−120∘=60∘\angle ABC = 180^\circ - 120^\circ = 60^\circ. Because all sides of a rhombus are equal in length, AB=BCAB = BC, making △ABC\triangle ABC an isosceles triangle with a 60∘60^\circ vertex angle, which implies △ABC\triangle ABC is equilateral. Therefore, side length AB=AC=12AB = AC = 12 inches, and the perimeter is 4×12=484 \times 12 = 48 inches.

Adım Adım Çözüm

1
Determine the consecutive angle measure in the rhombus.
\angle ABC = 180^\circ - 120^\circ = 60^\circ
Consecutive interior angles in a rhombus (which is a parallelogram) are supplementary.
2
Determine the nature of triangle ABC.
\triangle ABC is equilateral with AB = BC = AC = 12 inches.
A rhombus has four equal sides, so AB = BC. An isosceles triangle with a vertex angle of 60 degrees is an equilateral triangle.
3
Calculate the perimeter of the rhombus.
Perimeter = 4 \times 12 = 48 inches.
The perimeter of a rhombus is four times the length of one side.

Anahtar Kavram

Properties of a rhombus: all four sides are congruent, consecutive interior angles are supplementary, and a diagonal splitting a 60-degree angle forms an equilateral triangle with adjacent sides.
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Konular

Properties of Quadrilaterals

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