Question

The diagonals of a rhombus are \[16cm\] and \[12cm\] respectively. What is the measure of its side?
a). \[10cm\]
b). \[12cm\]
c). \[8cm\]
d). \[16cm\]

Answer 1

Hint: By the property of the rhombus, we know that all its sides are equal and that the diagonals are perpendicular and are at \[90\] degrees. So, it is very trivial that we can use the Pythagoras theorem with the help of the given diagonals to find its side.
Pythagoras theorem: \[{p^2} + {b^2} = {h^2}\], where \[p\] is the perpendicular, \[b\] is the base, and \[h\] is the hypotenuse.

Complete step-by-step solution:

Let ABCD be our rhombus where\[AC = 16cm\] and \[BD = 12cm\] as given to us and O be the intersection of the diagonals.
Let us consider the triangle\[AOB\] where \[AO = \dfrac{{AC}}{2} = 8cm\] and \[BO = \dfrac{{BD}}{2} = 6cm\].
Now we use the formula of Pythagoras theorem to find the side \[AB\]in the triangle\[AOB\]i.e.
\[{(AO)^2} + {(BO)^2} = {(AB)^2}\]
\[ \Rightarrow {8^2} + {6^2} = {(AB)^2}\]
\[ \Rightarrow {(AB)^2} = 64 + 36 = 100\]
\[ \Rightarrow {(AB)^2} = 100\]
\[ \Rightarrow AB = \sqrt {100} \]
\[ \Rightarrow AB = 10cm\]
Therefore, the length of the side of the rhombus is equal \[10cm\].
Hence, option (a) is correct.
Additional information: In geometry, a rhombus is a quadrilateral whose four sides all have the same length. since equilateral means that all of its sides are equal in length. The rhombus has two diagonals connecting pairs of opposite vertices, and two pairs of parallel sides. Opposite angles of a rhombus have equal measure. The two diagonals of a rhombus are perpendicular and bisect opposite angles.

Note: It is important that we know the basic geometric properties of the rhombus; we must know that the diagonals of the rhombus are at \[90\]degrees that give us the luxury to use the properties of triangles and let us compute the side of the rhombus using Pythagoras theorem.