Q: A rectangular park is 45 m long and 30 m wide. A path 2.5 m wide is constructed outside the park. Find the area of the path.

Area of park =l \times b= 45 \times 30=1350\ cm^2 Area of park with path is =45+2.5+2.5 \times 30+2.5+2.5 = 1750\ cm^2 Therefore,Area of path = Area of park with path - Area of rectangular park = 1750-1350=400\ cm^2 Hence, this is the answer.

Question 1

A rectangular park is 45 m long and 30 m wide. A path 2.5 m wide is constructed outside the park. Find the area of the path.

Accepted Answer

Area of park  =l 	imes b= 45 	imes 30=1350\ cm^2 Area of park with path is  =45+2.5+2.5 	imes 30+2.5+2.5 = 1750\ cm^2 Therefore,Area of path  =  Area of park with path  -  Area of rectangular park              = 1750-1350=400\ cm^2 Hence, this is the answer.

Question 2

The diagonals  AC  and  BD  of a parallelogram  ABCD  intersect each other at the point  O .If  \angle DAC = 32^o  and  \angle AOB = 70^o , then  \angle DBC  is equal to?

Accepted Answer

In given figure,Quadrilateral ABCD is a parallelogram.So, AD  \left|ight|  BC 	herefore   \angle DAC =  \angle ACB  --- ( Alternate angle) 	herefore   \angle ACB = 32 ^\circ \angle AOB +  \angle BOC = 180 ^\circ   --- (straight angle) \Rightarrow 70 ^\circ  +  \angle BOC = 180 ^\circ 	herefore   \angle BOC = 110 ^\circ In  	riangle BOC, \angle OBC +  \angle BOC +  \angle OCB = 180 ^\circ \Rightarrow \angle OBC  + 110 ^\circ  + 32 ^\circ  = 180 ^\circ \Rightarrow   \angle OBC = 38 ^\circ 	herefore    \angle DBC = 38 ^\circ

Question 3

Diagonals AC and BD of a parallelogram ABCD intersect each other at O. If  OA=3 cm and  OD=2 cm determine the lengths of AC and BD.

Accepted Answer

Here,  ABCD  is a parallelogram and  AC  and  BD  are diagonals of parallelogram intersect each other at  O. OA=3\,cm  and  OD=2\,cm We know that, diagonals of a parallelogram bisect each other. 	herefore    AO=OC  and  BO=OD \Rightarrow    AC=2	imes OA=2	imes 3\,cm=6\,cm \Rightarrow    BD=2	imes OD=2	imes 2\,cm=4\,cm 	herefore   The length of the diagonals  AC  and  BD  are  6\,cm  and  4\,cm.

Question 4

ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rhombus.

Accepted Answer

Here, we are joining A and C.In  \Delta ABCP is the mid point of ABQ is the mid point of BCPQ || AC [Line segments joining the mid points of two sides of a triangle is parallel to AC(third side) and also is half of it]PQ =\dfrac{1}{2} ACIn  \Delta ADCR is mid point of CDS is mid point of ADRS || AC [Line segments joining the mid points of two sides of a triangle is parallel to third side and also is half of it]RS =\dfrac{1}{2} ACSo, PQ || RS and PQ = RS [one pair of opposite side is parallel and equal]In  \Delta APS &  \Delta BPQAP = BP [P is the mid point of AB) \angle PAS =\angle PBQ(All the angles of rectangle are  90^o )AS = BQ 	herefore \Delta APS \cong\Delta BPQ(SAS congruency) 	herefore PS = PQBS = PQ & PQ = RS (opposite sides of parallelogram is equal) 	herefore  PQ = RS = PS = RQ[All sides are equal] 	herefore  PQRS is a parallelogram with all sides equal 	herefore  So PQRS is a rhombus.

Question 5

Show that a diagonal of a parallelogram divides into two congruent triangles and hence prove that the opposite sides of a parallelogram are equal.

Accepted Answer

REF. Image.consider  \Delta   ABC and  \Delta   ACDSince the line segments AB+CD are parallelto each other and AC is a transversal \angle   ACB =  \angle   CAD.AC = AC (common side) \angle  CAB =  \angle   ACD.Thus, by ASA criteria  \Delta  ABC  \cong     \Delta  ACDThe corresponding part of the congruenttriangle are congruentAB = CD + AD = BC

Question 6

The measures of two adjacent angles of a parallelogram are in the ratio  3 : 2 . Find the measure of each of the angles of the parallelogram

Accepted Answer

Let two adjacent angles A and B of  \parallel gm  ABCD be 3x and 2x respectively.Since the adjacent angles of a parallelogram are supplementary. \angle A+\angle B=180^o \Rightarrow 3x+2x=180^o \Rightarrow 5x=180^o \Rightarrow x=\dfrac {180^o}{5}=36^o 	herefore \angle 3	imes 36^o=108^o and,  \angle B=2	imes 36^o=72^o Since the opposite angles are equal in a parallelgram, therefore,  \angle C=\angle A=108^o  and  \angle D=\angle B=72^o Hence,  \angle A=108^o, \angle B=72^o, \angle C=108^o  and  \angle D=72^o

Question 7

Through the mid point M of the side CD of a parallelogram ABCD, the line BM is drawn intersecting AC at L and AD produced at E. Prove EL=2BL.

Accepted Answer

In  \bigtriangleup BMC  and   \bigtriangleup EMD \angle BMC=\angle EMD   [Vertically opposite] MC=DM  [Given] \angle BCM=\angle EDM  [Alternate angles] 	herefore \bigtriangleup  BMC\cong \bigtriangleup  EMD  [By ASA]Hence,  BC=DE  [By CPCT]  ightarrow (1) AE=AD+DE=BC+BC=2BC   ightarrow (2) Now,   \bigtriangleup BLC\sim \bigtriangleup ELA  (AA Similarly) \dfrac{BL}{EL}=\dfrac{BC}{AE}    [By CPCT] \dfrac{BL}{EL}=\dfrac{BC}{2BC} \dfrac{BL}{EL}=\dfrac{1}{2} EL=2BL

Question 8

State whether True or False.(a) All rectangles are squares.(b) All rhombuses are parallelograms.(c) All squares are rhombuses and also rectangles.(d) All squares are not parallelograms.(e) All kites are rhombuses.(f) All rhombuses are kites.(g) All parallelograms are trapeziums.(h) All squares are trapeziums.

Accepted Answer

(a) FalseAll squares are rectangles. (All sides of rectangle are not equal).(b) TrueOpposite sides are equal and parallel.(c) TrueAll sides of the rhombus are equal  	herefore  All squares are rhombuses. All squares are also rectangles as each interval angle measure  90^{\circ} (d) FalseAll squares are parallelogram as opposite sides are equal and parallel.(e) FalseKite has different lengths.(f) TrueRhombus also has two distinct consecutive pairs of sides of equal length.(g) TrueParallelogram pair of parallel sides.(h) TrueAll squares have a pair of parallel sides.

Question 9

The diagonals of a rhombus measure  16 cm and  30 cm. Find its perimeter.

Accepted Answer

Given: Diagonals AC =30 cm and DB =16 cm.Since the diagonals of the rhombus bisect at right angle to each other.Therefore, OD =\dfrac{DB}{2}=\dfrac{16}{2}=8 cmand OC =\dfrac{AC}{2}=\dfrac{30}{2}=15 cmNow, In right angle triangle DOC, (DC)^2=(OD)^2+(CO)^2 \Rightarrow (DC)^2=(8)^2+(15)^2 \Rightarrow (DC)^2=64+225=289 \Rightarrow DC=\sqrt{289}=17 cmPerimeter of the rhombus =4	imes  side =4	imes 17=68 cmThus, the perimeter of rhombus is  68  cm.

Question 10

How many sides does a pentagon have?

Accepted Answer

A polygon has as many angles as sides.Like, triangle has  3  angles and  3  sides.A pentagon means a polygon which has  5  sides and  5  angles.Hence, option B is correct.