Question

Choose the correct alternative answer and fill in the blanks.

(i) If all pairs of adjacent sides of a quadrilateral are congruent then it is called ....

(A) rectangle (B) parallelogram (C) trapezium, (D) rhombus

(ii) If the diagonal of a square is 12 $\sqrt{2}$ cm then the perimeter of square is ......

(A) 24 cm (B) 24 $\sqrt{2}$ cm (C) 48 cm (D) 48 $\sqrt{2}$ cm

(iii) If opposite angles of a rhombus are (2x)^° and (3x $-$ 40)^° then value of x is ...

(A) 100^° (B) 80^° (C) 160^° (D) 40^°

Answer 1

(i) When all the pairs of the adjacent sides of a quadrilateral are congruent then it is a rhombus.
Hence, the correct answer is option D.

(ii)

Let the diagonal be AC = 12$\sqrt{2}$ cm
since it is a squarem so, all the angles are 90$°$.
In $△$ABC, we apply the Pythagoras theorem,
$$\begin{gathered} {\mathrm{AB}}^2+{\mathrm{BC}}^2={\mathrm{AC}}^2 \\ \Rightarrow {\mathrm{AB}}^2+{\mathrm{AB}}^2={\mathrm{AC}}^2\left(\because \mathrm{ABCD}\mathrm{is}\mathrm{a}\mathrm{square}\right) \\ \Rightarrow 2{\mathrm{AB}}^2={\mathrm{AC}}^2 \\ \Rightarrow 2{\mathrm{AB}}^2={\left(12\sqrt{2}\right)}^2 \\ \Rightarrow 2{\mathrm{AB}}^2=288 \\ \Rightarrow {\mathrm{AB}}^2=144 \\ \Rightarrow \mathrm{AB}=12\mathrm{cm} \end{gathered}$$
So, AB = BC = CD = AD = 12 cm
Perimeter of the square ABCD = 4AB = $4\times 12\mathrm{cm}=48\mathrm{cm}$
Hence, the correct answer is option C.

(iii) A rhombus is also a parallelogram so, the opposite angles will be congruent.
Thus, (2x)^° = (3x $-$ 40)°
$$\begin{gathered} \Rightarrow 3x-2x=40° \\ \Rightarrow x=40° \end{gathered}$$
Hence, the correct answer is option D.