Exercise 2.2 - Chapter 2 Measurements Term 1 7th Maths Guide Samacheer Kalvi Solutions - SaraNextGen [2024-2025]

Updated By SaraNextGen

On April 24, 2024, 11:35 AM

Ex $2.2$
Question $1 .$
Find the area of rhombus PQRS shown in the following figures.

Solution:
(i) Given the diagonals $\mathrm{d}_{1}=16 \mathrm{~cm} ; \mathrm{d}_{2}=8 \mathrm{~cm}$
Area of the rhombus $=\frac{1}{2}\left(\mathrm{~d}_{1} \times \mathrm{d}_{2}\right)$ sq. units
$=\frac{1}{2} \times 16 \times 8 \mathrm{~cm}^{2}=64 \mathrm{~cm}^{2}$
Area of the rhombus $=64 \mathrm{~cm}^{2}$
(ii) Given base $\mathrm{b}=15 \mathrm{~cm}$; Height $\mathrm{h}=11 \mathrm{~cm}$
Area of the rhombus $=$ (base $\times$ height) sq. units
$=15 \times 11 \mathrm{~cm}^{2}=165 \mathrm{~cm}^{2}$
Area of the rhombus $=165 \mathrm{~cm}^{2}$

Question $2 .$
Find the area of a rhombus whose base is $14 \mathrm{~cm}$ and height is $9 \mathrm{~cm}$.
Solution:

Given base $\mathrm{b}=14 \mathrm{~cm}$; Height $\mathrm{h}=9 \mathrm{~cm}$ Area of the rhombus $=\mathrm{b} \times \mathrm{h} \mathrm{sq}$. units $=14 \times 9 \mathrm{~cm}^{2}=126 \mathrm{~cm}^{2}$

Question $3 .$
Find the missing value.

Solution:
(i) Given diagonal $\mathrm{d}_{1}=19 \mathrm{~cm} ; \mathrm{d}_{2}=16 \mathrm{~cm}$
Area of the rhombus $=\frac{1}{2}\left(\mathrm{~d}_{1} \times \mathrm{d}_{2}\right)$ sq. units $=\frac{1}{2} \times 19 \times 16$ $=152 \mathrm{~cm}^{2}$
(ii) Given diagonal $\mathrm{d}_{1}=26 \mathrm{~m}$; Area of the rhombus $=468$ sq. $\mathrm{m}$ $=\frac{1}{2}\left(\mathrm{~d}_{1} \times \mathrm{d}_{2}\right)=468 ;\left(26 \times \mathrm{d}_{2}\right)=468 \times 2$ $\mathrm{d}_{2}=\frac{468 \times 2}{26}=\mathrm{d}_{2}=36 \mathrm{~m}$
(iii) Given diagonal $\mathrm{d}_{2}=12 \mathrm{~mm}$; Area of the rhombus $=180 \mathrm{sq} . \mathrm{m}$ $\frac{1}{2}\left(\mathrm{~d}_{1} \times \mathrm{d}_{2}\right)=180$
$\begin{aligned}
&\frac{1}{2}\left(\mathrm{~d}_{1} \times 12\right)=180 \\
&\mathrm{~d}_{1} \times 12=180 \times 2 \\
&\mathrm{~d}_{1}=\frac{180 \times 2}{12} \\
&\mathrm{~d}_{1}=30 \mathrm{~mm}
\end{aligned}$
$\text { Diagonal } \mathrm{d}_{1}=30$
Diagonal $\mathrm{d}_{1}=30 \mathrm{~mm}$

Question $4 .$
The area of a rhombus is 100 sq. $\mathrm{cm}$ and length of one of its diagonals is $8 \mathrm{~cm}$. Find the length of the other diagonal.
Solution:
Given the length of one diagonal $\mathrm{d}_{1}=8 \mathrm{~cm}$; Area of the rhombus $=100 \mathrm{sq} . \mathrm{cm}$
$\frac{1}{2}\left(\mathrm{~d}_{1} \times \mathrm{d}_{2}\right)=100$
$\frac{1}{2} \times 8 \times \mathrm{d}_{2}=100$
$8 \times \mathrm{d}_{2}=100 \times 2$ $\mathrm{~d}_{2}=100 \times 2=25 \mathrm{~cm}$
Length of the other diagonal $\mathrm{d}_{2}=25 \mathrm{~cm}$

Question 5 .
A sweet is in the shape of rhombus whose diagonals are given as $4 \mathrm{~cm}$ and $5 \mathrm{~cm}$. The surface of the sweet should be covered by an aluminum foil. Find the cost of aluminum foil used for 400 such sweets at the rate of $₹ 7$ per 100 sq. $\mathrm{cm}$.
Solution:
Diagonals $\mathrm{d}_{1}=4 \mathrm{~cm}$ and $\mathrm{d}_{2}=5 \mathrm{~cm}$
Area of one rhombus shaped sweet $=\frac{1}{2}\left(\mathrm{~d}_{1} \times \mathrm{d}_{2}\right)$ sq. units $=\frac{1}{2} \times 4 \times 5 \mathrm{~cm}^{2}=10 \mathrm{~cm}^{2}$
Aluminum foil used to cover 1 sweet $=10 \mathrm{~cm}^{2}$
$\therefore$ Aluminum foil used to cover 400 sweets $=400 \times 10=4000 \mathrm{~cm}^{2}$
Cost of aluminum foil for $100 \mathrm{~cm}^{2}=₹ 7$
$\therefore$ Cost of aluminum foil for $4000 \mathrm{~cm}^{2}=\frac{4000}{100} \times 7=₹ 280$
$\therefore$ Cost of aluminum foil used $=₹ 280$.

Question $6 .$
The area of the rhombus with side $4 \mathrm{~cm}$ and height $3 \mathrm{~cm}$ is
(i) 7 sq. $\mathrm{cm}$
(ii) $24 \mathrm{sq} . \mathrm{cm}$
(iii) 12 sq. $\mathrm{cm}$
(iv) 10 sq. $\mathrm{cm}$
Solution:
(iii) $12 \mathrm{sq} . \mathrm{cm}$
Hint:
Area $=$ Base $\times$ Height $=4 \times 3=12 \mathrm{~cm}^{2}$
 

Question $7 .$
The area of the rhombus when both diagonals measuring $8 \mathrm{~cm}$ is
(i) 64 sq. $\mathrm{cm}$
(ii) $32 \mathrm{sq} . \mathrm{cm}$
(iii) 30 sq. $\mathrm{cm}$
(iv) $16 \mathrm{sq} . \mathrm{cm}$
Solution:
(ii) $32 \mathrm{sq} . \mathrm{cm}$
Hint:
Area $=\frac{1}{2}\left(\mathrm{~d}_{1} \times \mathrm{d}_{2}\right)=\frac{1}{2} \times 8 \times 8=32$

Question $8 .$
The area of the rhombus is 128 sq. $\mathrm{cm}$. and the length of one diagonal is $32 \mathrm{~cm}$. The length of the other diagonal is
(i) $12 \mathrm{~cm}$
(ii) $8 \mathrm{~cm}$
(iii) $4 \mathrm{~cm}$
(iv) $20 \mathrm{~cm}$
Solution:
(ii) $8 \mathrm{~cm}$
Hint:
$\frac{1}{2} \times \mathrm{d}_{1} \times \mathrm{d}_{2}=128 \Rightarrow \mathrm{d}_{2}=\frac{128 \times 2}{32}=8 \mathrm{~cm}$
Question $9 .$
The height of the rhombus whose area 96 sq. $\mathrm{m}$ and side $24 \mathrm{~m}$ is
(i) $8 \mathrm{~m}$
(ii) $10 \mathrm{~m}$
(iii) $2 \mathrm{~m}$
(iv) $4 \mathrm{~m}$
Solution:
(iv) $4 \mathrm{~m}$

Question $10 .$
The angle between the diagonals of a rhombus is
(i) $120^{\circ}$
(ii) $180^{\circ}$
(iii) $90^{\circ}$
(iv) $100^{\circ}$
Solution:
(iii) $90^{\circ}$
Hint:
Angles of a rhombus bisect at right angles.