Question

Lengths of diagonals of a rhombus ABCD are 16 cm and 12 cm. Find the side and perimeter of the rhombus.

Answer 1

ABCD is a rhombus.

Here, segment AC and segment BD are the diagonals of the rhombus ABCD.

l(AC) = 16 cm and l(BD) = 12 cm.

Diagonals of a rhombus are perpendicular bisectors of each other.

∴ m∠AOD = 90°

Also, l(OA) = ${\displaystyle \frac{1}{2}l\left(AC\right)=\frac{1}{2}\times 16=8 cm}$

l(OD) = ${\displaystyle \frac{1}{2}l\left(BD\right)=\frac{1}{2}}$× 12 = 6 cm

In right ∆AOD, 

l(AD)² = l(OA)² + l(OD)²               ...[Pythagoras theorem]

⇒ l(AD)² = (8)² + (6)² 

⇒ l(AD)² = 64 + 36 = 100

⇒ l(AD) = $$\sqrt{100}$$ = 10 cm

All sides of a rhombus are equal.

∴ Perimeter of the rhombus ABCD = 4 × Side of a rhombus = 4 × 10 = 40 cm

Thus, the side and perimeter of the rhombus are 10 cm and 40 cm, respectively.