Note: Figure is not drawn to scale.
If x = 4 units, y = 6 units, and h = 5 units, find the area of the rhombus shown above using decomposition.
A.
15 square units
B.
30 square units
C.
120 square units
D.
50 square units
To find the area of the rhombus using decomposition, we first need to decompose the rhombus into two congruent triangles by drawing a line from one vertex to the midpoint of the opposite side. This creates two right triangles with legs x/2 and y/2, and hypotenuse h.
Using the Pythagorean theorem, we can find the length of the legs of the triangles:
(x/2)^2 + (y/2)^2 = h^2
(4/2)^2 + (6/2)^2 = 5^2
2^2 + 3^2 = 25
4 + 9 = 25
13 = 25
Now we can find the area of one of the triangles using the formula for the area of a triangle:
Area = 1/2 * base * height
Area = 1/2 * 4 * 3
Area = 6 square units
Since the rhombus is composed of two congruent triangles, the total area of the rhombus is twice the area of one triangle:
Total area = 2 * 6
Total area = 12 square units
Therefore, the correct answer is A. 15 square units.