Rhombus - MGMAT

singhsa wrote:In the rhombus ABCD, the length of diagonal BD is 6 and the length of diagonal AC is 8. What is the perimeter of ABCD?

a. 10
b. 14
c. 20
d. 24
e. 28

OA - c

We know that the two diagnols of a rhombus bisect each other and form a 90 degree angle at the intersection. So if you draw a rhombus with intersecting diagnols, you get 4 triangles. Since you know each diagnol is cut into half by the other diagnol, you have the 2 sides of the triangle. Using pyathorean theorum, calculate the hypotenuse using sides 3 and 4, you get 5.

Perimeter = 4(5) = 20 (C)

Agree the ans is C

A rhombus is a parallelogram with four sides of equal length. Thus, AB = BC = CD = DA.

The diagonals of a parallelogram bisect each other, meaning that AC and BD intersect at their midpoints, which we will call E. Thus, AE = EC = 4 and BE = ED = 3. Since ABCD is a rhombus, diagonals AC and BD are also perpendicular to each other.

Labeling the figure with the lengths above, we can see that the rhombus is divided by the diagonals into four right triangles, each of which has one side of length 3 and another side of length 4.

Remembering the common right triangle with side ratio = 3: 4: 5, we can infer that the unlabeled hypotenuse of each of the four triangles has length 5.

Thus, AB = BC = CD = DA = 5, and the perimeter of ABCD is 5 × 4 = 20.

The correct answer is C.