Exercise 5.5 - Chapter 5 Geometry 8th Maths Guide Samacheer Kalvi Solutions - SaraNextGen [2024-2025]

Updated By SaraNextGen

On April 24, 2024, 11:35 AM

$\mathbf{E x} 5.5$
I. Construct the following parallelograms with the given measurements and find their area.
Question $1 .$
ARTS, $\mathrm{AR}=6 \mathrm{~cm}, \mathrm{RT}=5 \mathrm{~cm}$ and $\angle \mathrm{ART}=70^{\circ}$.
Answer:
Given : In the Parallelogram ARTS,
$\mathrm{AR}=6 \mathrm{~cm}, \mathrm{RT}=5 \mathrm{~cm} \text {, and } \angle \mathrm{ART}=70^{\circ}$

Construction:
Stcps:
- Draw a line segment $A R=6 \mathrm{~cm}$.
- Make an angle $\angle \mathrm{ART}=70^{\circ}$ at $\mathrm{R}$ on $\mathrm{AR}$
- With $\mathrm{R}$ as centre, draw an arc of radius $5 \mathrm{~cm}$ cutting $\mathrm{RX}$ at $\mathrm{T}$
- Draw a line TY parallel to AR through T.
- With T as centre, draw an arc of radius $6 \mathrm{~cm}$ cutting TY at S. Join AS
- ARTS is the required parallelogram.
Calculation of area:
Area of the parallelogram ARTS $=\mathrm{b} \times \mathrm{h}$ sq. units $=6 \times 4.7=28.2$ sq.cm

Question $2 .$
CAMP, $\mathrm{CA}=6 \mathrm{~cm}, \mathrm{AP}=8 \mathrm{~cm}$ and $\mathrm{CP}=5.5 \mathrm{~cm}$.
Answer:
Given : In the parallelogram CAMP,
$\mathrm{CA}=6 \mathrm{~cm}, \mathrm{AP}=8 \mathrm{~cm}$, and $\mathrm{CP}=5.5 \mathrm{~cm}$

Construction:
Steps:
- Draw a line segment $\mathrm{CA}=6 \mathrm{~cm}$.
- With C as centre, draw an arc of length $5.5 \mathrm{~cm}$
- With A as centre, draw an arc of length $8 \mathrm{~cm}$
- Mark the intersecting point of these two arcs as $P$
- Draw a line PX parallel to CA
- With P as centre draw an arc of radius $6 \mathrm{~cm}$ cutting PX at M. Join AM
- CAMP is the required parallelogram.
Calculation of area:
Area of the Parallelogram CAMP $=b \times h$ sq. units $=6 \times 5.5=33 \mathrm{sq} . \mathrm{cm}$
 

Question $3 .$
EARN, ER $=10 \mathrm{~cm}, \mathrm{AN}=7 \mathrm{~cm}$ and $\angle \mathrm{EOA}=110^{\circ}$ where $\overline{\mathrm{ER}}$ and $\overline{\mathrm{AN}}$ intersect at O.
Answer:
Given: In the parallelogram EARN,
$\mathrm{ER}=10 \mathrm{~cm}, \mathrm{AN}=7 \mathrm{~cm}$, and $\mathrm{LEOA}=1100$
Where $\overline{\mathrm{ER}}$ and $\overline{\mathrm{AN}}$ intersect at 0

Construction:
Steps:
- Draw a line segment PX. Mark a point $\mathrm{O}$ on $\mathrm{PX}$
- Make an angle $\angle \mathrm{EOA}=1100$ on PX at $\mathrm{O}$
- Draw arcs of radius $3.5 \mathrm{~cm}$ with $\mathrm{O}$ as centre on either side of PX. Cutting $\mathrm{YZ}$ on $A$ and $N$
- With $\mathrm{A}$ as centre, draw an arc of radius $10 \mathrm{~cm}$, cutting $\mathrm{PX}$ at $\mathrm{E}$. Join $\mathrm{AE}$
- Draw a line parallel to $\mathrm{AE}$ at $\mathrm{N}$ cutting $\mathrm{PX}$ at $\mathrm{R}$. Join $\mathrm{EN}$ and $\mathrm{AR}$
- EARN is the required parallelogram
Calculation of area:
Area of the Parallelogram EARN $=\mathrm{b} \times \mathrm{h}$ sq. units $=10 \times 5.5=55$ sq.cm
 

Question $4 .$
GAIN, GA $=7.5 \mathrm{~cm}, \mathrm{GI}=9 \mathrm{~cm}$ and $\angle \mathrm{GAI}=100^{\circ}$.
Answer:
Given : In the parallelogram GAIN,
$\mathrm{GA}=7.5 \mathrm{~cm}, \mathrm{GI}=9 \mathrm{~cm}$, and $\angle \mathrm{GAI}=100^{\circ}$

Construction:
Steps:
- Draw a line segment $\mathrm{GA}=7.5 \mathrm{~cm}$.
- Make an angle GAI $=100^{\circ}$ at $A$.
- With G as centre, draw an arc of radius $9 \mathrm{~cm}$ cutting AX at I. Join GI.
- Draw a line IY parallel to GA through I.
- With I as centre, draw an arc of radius $7.5 \mathrm{~cm}$ on IY cutting at N. Join GN
- GAIN is the required parallelogram.
Construction of area:
Area of the Parallelogram GAIN $=\mathrm{b} \times \mathrm{h}$ sq. units $=7.5 \times 39=29.25 \mathrm{sq} . \mathrm{cm}$

II. Construct the following rhombuses with the given measurements and also find their area.
(i) $\mathrm{FACE}, \mathrm{FA}=6 \mathrm{~cm}$ and $\mathrm{FC}=8 \mathrm{~cm}$
Answer:
Given $\mathrm{FA}=6 \mathrm{~cm}$ and $\mathrm{FC}=8 \mathrm{~cm}$

Steps:
- Drawn a line segment FA $=6 \mathrm{~cm}$.
- With $\mathrm{F}$ and $\mathrm{A}$ as centres, drawn arcs of radii $8 \mathrm{~cm}$ and $6 \mathrm{~cm}$ respectively and let them cut at $C$.
- Joined FC and AC.
- With $\mathrm{F}$ and $\mathrm{C}$ as centres, drawn arcs of radius $6 \mathrm{~cm}$ each and let them cut at $\mathrm{E}$.
- Joined FE and EC.
- FACE is the required rhombus.
Calculation of Area :
Area of the rhombus $=\frac{1}{2} \times \mathrm{d}_{1} \times \mathrm{d}_{2}$ sq.units $=\frac{1}{2} \times 8 \times 9$ sq.units $=36 \mathrm{~cm}$
(ii) $\mathrm{CAKE}, \mathrm{CA}=5 \mathrm{~cm}$ and $\angle \mathrm{A}=65^{\circ}$
Answer:
Given $\mathrm{CA}=5 \mathrm{~cm}$ and $\angle \mathrm{A}=65^{\circ}$

Steps:
- Drawn a line segment $\mathrm{CA}=5 \mathrm{~cm}$.
- At $\mathrm{A}$ on $\mathrm{AC}$, made $\angle \mathrm{CAX}=65^{\circ}$
- With A as centre, drawn arc of radius $5 \mathrm{~cm}$. Let it cut $\mathrm{AX}$ at $\mathrm{K}$.
- With $\mathrm{K}$ and $\mathrm{C}$ as centres, drawn arcs of radius $5 \mathrm{~cm}$ each and let them cut at E.
- Joined KE and CE.
- CAKE is the required rhombus.
Calculation of Area:
Area of the rhombus $=\frac{1}{2} \times \mathrm{d}_{1} \times \mathrm{d}_{2}$ sq.units
$\begin{aligned}
&=\frac{1}{2} \times 54 \times 85 \mathrm{~cm}^{2} \\
&=22.95 \mathrm{~cm}^{2}
\end{aligned}$
(iii) $\mathrm{LUCK}, \mathrm{LC}=7.8 \mathrm{~cm}$ and $\mathrm{UK}=6 \mathrm{~cm}$
Answer:
Given $\mathrm{LC}=7.8 \mathrm{~cm}$ and $\mathrm{UK}=6 \mathrm{~cm}$

Steps:
- Drawn a line segment $\mathrm{LC}=7.8 \mathrm{~cm}$.
- Drawn the perpendicular bisector $\mathrm{XY}$ to $\mathrm{LC}$. Let it cut $\mathrm{LC}$ at ' $\mathrm{O}$ '
- With $\mathrm{O}$ as centres, drawn arc of radius $3 \mathrm{~cm}$ on either side of $\mathrm{O}$ which cut $\mathrm{OX}$ at $\mathrm{K}$ and $\mathrm{OY}$ at $\mathrm{U}$.
- Joined LU, UC, CK and LK.
- UCK is the required rhombus.
Calculation of Area:
Area of the rhombus $=\frac{1}{2} \times \mathrm{d}_{1} \times \mathrm{d}_{2}$ sq.units
$=\frac{1}{2} \times 7.8 \times 6 \mathrm{~cm}^{2}=23.4 \mathrm{~cm}^{2}$
(iv) $\mathrm{PARK}, \mathrm{PR}=9 \mathrm{~cm}$ and $\angle \mathrm{P}=70^{\circ}$
Answer:
Given $\mathrm{PR}=9 \mathrm{~cm}$ and $\angle \mathrm{P}=70^{\circ}$

Steps:
- Drawn a line segment $P R=9 \mathrm{~cm}$.
- At $\mathrm{P}$, made $\angle R P X \angle R P Y=35^{\circ}$ on either side of $P R$.
- At R, made $\angle \mathrm{PRQ}=\angle \mathrm{PRS}=35^{\circ}$ on either side of $\mathrm{PR}$
- Let PX and RQ cut at A and PY and RS at K.
- PARK is the required rhombus
Constructon of Area:
Area of the rhombus $=\frac{1}{2} \times \mathrm{d}_{1} \times \mathrm{d}_{2}$ sq.units $=\frac{1}{2} \times 9 \times 6.2 \mathrm{~cm}^{2}$ $=27.9 \mathrm{~cm}^{2}$
III. Construct the following rectangles with the given measurements and also find their area.
(i) $\mathrm{HAND}, \mathrm{HA}=7 \mathrm{~cm}$ and $\mathrm{AN}=4 \mathrm{~cm}$
Answer:
Given $\mathrm{HA}=7 \mathrm{~cm}$ and $\mathrm{AN}=4 \mathrm{~cm}$

Steps:
- Drawn a line segment $\mathrm{HA}=7 \mathrm{~cm}$.
- At H, constructed HX $\perp \mathrm{HA}$.
- With $\mathrm{H}$ as centre, drawn an arc of radius $4 \mathrm{~cm}$ and let it cut at HX at D.
- With A and D as centres, drawn ares of radii $4 \mathrm{~cm}$ and $7 \mathrm{~cm}$ respectively and let them cut at $\mathrm{N}$.
- Joined AN and DN.
- HAND is the required rectangle.
calculation of' area :
Area of the rectangle HAND $=1 \times b$ sq.units
$\begin{aligned}
&=7 \times 4 \mathrm{~cm}^{2} \\
&=28 \mathrm{~cm}^{2}
\end{aligned}$
(ii) $\mathrm{LAND}, \mathrm{LA}=8 \mathrm{~cm}$ and $\mathrm{AD}=10 \mathrm{~cm}$
Answer:
Given $\mathrm{LA}=8 \mathrm{~cm}$ and $\mathrm{AD}=10 \mathrm{~cm}$

Sleps :
- Drawn a line segment $\mathrm{LA}=8 \mathrm{~cm}$.
- At L, constructed LX $\perp \mathrm{LA}$.
- With A as centre, drawn an arc of radius $10 \mathrm{~cm}$ and let it cut at LX at D.
- With A as centre and LD as radius drawn an arc. Also with D as centre and LA as radius drawn another arc. Let then cut at $\mathrm{N}$.
- Joined DN and AN.
- LAND is the required rectangle.
Calcualtion of arca :
Area of the rectangle $\mathrm{LAND}=1 \times \mathrm{b}$ sq.units
$\begin{aligned}
&=8 \times 5.8 \mathrm{~cm}^{2} \\
&=46.4 \mathrm{~cm}^{2}
\end{aligned}$
IV. Construct the following squares with the given measurements and also find their area.
(i) $\mathrm{EAST}, \mathrm{EA}=6.5 \mathrm{~cm}$
Answer:
Given side $=6.5 \mathrm{~cm}$

Steps:
- Drawn a line segment EA $=6.5 \mathrm{~cm}$.
- With E as centre, drawn an arc of radius $6.5 \mathrm{~cm}$ and let it cut EX at T.
- With A and T as centre drawn an arc of radius $6.5 \mathrm{~cm}$ each and let them cut at $\mathrm{S}$.
- Joined TS and AS.
- EAST is the required square.
Calcualtion of Area:
Area of the square EAST $=\mathrm{a}^{2}$ sq.units
$\begin{aligned}
&=6.5 \times 6.5 \mathrm{~cm}^{2} \\
&=42.25 \mathrm{~cm}^{2}
\end{aligned}$

Steps:
- Drawn a line segment WS $=7.5 \mathrm{~cm}$.
- Drawn the perpendicular bisector XY to WS. Let it bisect BS at O.
- With $\mathrm{O}$ as centre, drawn an arc of radius $3.7 \mathrm{~cm}$ on cither side of $\mathrm{O}$ which cut $\mathrm{OX}$ at $\mathrm{T}$ and $\mathrm{OY}$ at $\mathrm{E}$
- Joined BE, ES, ST and BT.
- WEST is the required square.
Calculation of Area:
Area of the square WEST $=\mathrm{a}^{2}$ sq.units
$\begin{aligned}
&=5.3 \times 53 \mathrm{~cm}^{2} \\
&=28.09 \mathrm{~cm}^{2}
\end{aligned}$