Case study (4 Marks)

Question 14 Marks

Two new roads, Road $E$ and Road $F$ were constructed between society $4$ and $1$ and society $1$ and $2.$

$5.$ What would be the measure of the sum of angles formed by the straight roads at society $1$ and society $3?$
$A. 60^\circ $
$B. 90^\circ $
$C. 180^\circ $
$D. 360^\circ $
$6.$ Krish says, “The distance to go from society $4$ to society $2$ using Road $D$ will be longer that the distance using Road $E”$
Is Krish correct$?$ Justify your answer with examples.
$7.$ Road $G,$ perpendicular to Road $F$ was constructed to connect the park and Road $F.$
Which of the following is true for Road $G$ and Road $F?$
$A.$ Road $G$ and road $F$ are of same length.
$B$. Road $F$ divides Road $G$ into two equal parts.
$C.$ Road $G$ divides Road $F$ into two equal parts.
$D.$ The length of road $G$ is one-fourth of the length of Road $F.$
$8.$ Priya said, “Minor arc corresponding to Road $B$ is congruent to minor arc corresponding to Road $D.”$
Do you agree with Priya$?$ Give reason to support your answer.

Answer

$5. C. 180°$
$6.$ Examples to show that in a right triangle the sum of legs is longest for an isosceles right triangle when hypotenuse remains same.
● Take for example the length of diameter $($hypotenuse$) = 5$ units.
Road $D$ and Road $B$ are equal hence $($Road $D = 3.53$ units$).$
Let Road $E$ be $= 1$ Chapter, Road $F = 4.89$ units.
Therefore, length of Road $B\ +$ Road $D$ is greater than Road $E\ +$ Road $F.$
$7. C.$ Road G divides Road F into two equal parts.
$8.$ Yes, Priya is correct with valid reasoning.
● Yes, Priya is correct because arc corresponding to two equal roads (chords) are congruent.

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Question 24 Marks

Given below is the map giving the position of four housing societies in a township connected by a circular road $A.$

Society $2$ and $3$ are connected by straight road $B,$ society $4$ and $2$ are connected by straight road $C$ and society $ 4$ and 3 are connected by road $D.$ Point $P$ denotes the position of a park. The park is equidistant to all four societies.
Rubina claims that it is not possible to construct another circular road connecting all four societies.
$1.$ Which of the following options justiies Rubina’s claim?
$A.$ Equal chords of congruent circles subtend equal angles at the centre.
$B.$ The perpendicular from the centre of a circle to a chord bisects the chord.
$C.$ There is a unique circle passing through three non-collinear points.
$D.$ Points equidistant from a given point will lie on a circle.
$2$ What is the position of the park P with respect to road $A?$
$A.$ Chord
$B.$ Centre
$C.$ Sector
$D.$ Segment
$3.$ The length of Road $B$ is equal to the length of Road $D.$
Which of the following options can be true for the roads in the township$?$
$A.$ Road $B$ bisects Road $D.$
$B.$ Road $B$ and Road make an acute angle.
$C.$ Road $B,$ Road $C$ and Road $D$ are of equal length.
$D. $ Road $B$ and Road $D$ subtend equal angles at society $1.$
$4.$ Alex says, “The angle made by road $B$ on road $D$ is a right angle.”
Jai and Angad give different justiications to support Alex’s claim.
Jai says, “Angles in the same segment of a circle are equal.”
Angad says, “The angle in a semicircle is a right angle.”
Who has given the correct justiication$?$

Answer

$1. C. $ There is a unique circle passing through three non-collinear points.
$2. B.$ Centre
$3. D.$ Road $B$ and Road $D$ subtend equal angles at society $1.$
$4.$ Angad is correct.

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Question 34 Marks

Read the Source/ Text given below and answer these questions:

As Class $IX\ C'$ s teacher Mrs.Rashmi entered in the class, She told students to do some practice on circle chapter. She Draws two-line $AB$ and $BC$ so that $AB = 8\ cm$ and $BC = 6\ cm.$ She told all students To make this shape in their notebook and draw a circle passing through the three points $A, B$ and $C.$
$i.$ Dileep drew $AB$ and $BC$ as per the figure
$ii.$ He drew perpendicular bisectors $OP$ and $OQ$ of the line $AB$ and $BC.$
$ii. OP$ and $OQ$ intersect at $O$
$iv.$ Now taking $O$ as centre and $OB$ as radius he drew The circle which passes through $A, B$ and $C.$
$v.$ He noticed that $A, O$ and $C$ are collinear.
Answer the following questions:
$i.$ What you will call the line $\text{AOC}?$
$a.$ Arc
$b.$ Diameter
$c.$ Radius
$d.$ Chord
$ii.$ What is the measure of $\angle\text{ABC}?$
$a. 60^\circ $
$b. 90^\circ $
$c. 45^\circ $
$d. 75^\circ $
$iii.$ What you will call the yellow color shaded area $\text{AMB}?$
$a.$ Arc.
$b.$ Sector.
$c.$ Major segment.
$d.$ Minor Segment.
$iv.$ What you will call the grey colour shaded area $\text{BCNA}?$
$a.$ Arc.
$b.$ Sector.
$c.$ Major segment.
$d.$ Minor Segment.
$v.$ What is the radius of the circle$?$
$a. 4\ cm$
$b. 3\ cm$
$c. 7\ cm$
$d. 5\ cm$

Answer

$(i)$	$(b)$	Diameter
$(ii)$	$(b)$	$90^\circ $
$(iii)$	$(d)$	Minor Segment.
$(iv)$	$(c)$	Major segment.
$(v)$	$(d)$	$5\ cm$

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Question 44 Marks

Read the Source/ Text given below and answer any four questions: Rohan and Suraj were close friends, One day they were riding horses from Delhi to Faridabad. The names of their horses were Saku and Fareed respectively. The day was very sunny. On the way, they stopped for resting in a park. They tied their horses to a tree in the park. The length of ropes of Rohans's horse is $14m$ and that of the horse of Suraj is $7m$ as shown in the figures. Both the friends slept in the park under a green tree for some time. During this period both the horses took $10$ rounds along with the tree they were tied.

Answer the following questions
$i.$ The ratio of distance walked in $10$ rounds by the horses of Rohan and Suraj is:
$a. 2 : 1$
$b. 1 : 2$
$c. 3 : 1$
$d. 1 : 3$
$ii.$ The ratio of area of the grass the horses of Rohan and Suraj could graze:
$a. 2 : 1$
$b. 1 : 2$
$c. 4 : 1$
$d. 1 : 4$
$iii.$ What is the distance walked by Rohan's horse in $5$ rounds:
$a. 220\ m$
$b. 100\ m$
$c. 440\ m$
$d. 110\ m$
$iv.$ What we call the the length of rope in terms of circle$?$
$a.$ Diameter
$b.$ Radius
$c.$ Chord
$d.$ Tangent
$v.$ What we call the the distance walked by a horse in one round$?$
$a.$ Area
$b.$ Radius
$c.$ Circumference
$d.$ diameter

Answer

$i$	$a$	$2 : 1$
$ii$	$c$	$4 : 1$
$iii$	$c$	$440\ m$
$iv$	$b$	Radius
$v$	$c$	Circumference

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Question 54 Marks

Read the Source/ Text given below and answer these questions:

A farmer has a circular garden as shown in the picture above. He has a different type of trees, plants and flower plants in his garden. In the garden, there are two mango trees $A$ and $B$ at a distance of $AB = 10m.$ Similarly, he has two Ashoka trees at the same distance of $10\ m$ as shown at $C$ and $D.\ AB$ subtends $\angle\text{AOB}=120^\circ$ at the center $O,$ The perpendicular distance of $AC$ from center is $5m$. The radius of the circle is $13\ m$. Now answer the following questions:
$i.$ What is the value of $\angle\text{COD}?$
$a. 60^\circ $
$b. 120^\circ $
$c. 100^\circ $
$d. 80^\circ $
$ii.$ What is the distance between mango tree $A$ and Ashok tree $C?$
$a. 12\ m$
$b. 24\ m$
$c. 13\ m$
$d. 15\ m$
$iii.$ What is the value of $\angle\text{OAB}?$
$a. 60^\circ $
$b. 120^\circ $
$c. 30^\circ $
$d. 90^\circ $
$iv.$ What is the value of $\angle\text{OCD?}$
$a. 30^\circ $
$b. 120^\circ $
$c. 60^\circ $
$d. 90^\circ $
$v.$ What is the value of $\angle\text{ODC}?$
$a. 90^\circ $
$b. 120^\circ $
$c. 60^\circ $
$d. 30^\circ $

Answer

$(i)$	$(b)$	$120^\circ $
$(ii)$	$(b)$	$24m$
$(iii)$	$(c)$	$30^\circ $
$(iv)$	$(a)$	$30^\circ $
$(v)$	$(d)$	$30^\circ $

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Question 64 Marks

Read the Source Text given below and answer these questions:

There was a circular park in Defence colony At Delhi. For fencing purpose Poles $A, B, C$ and $D$ were installed at the circumference of the park. Ram tied wires From $A$ to $B$ to $C$ and $C$ to $D,$ He managed to measure the $\angle\text{A}=100^\circ$ and $\angle\text{D}=80^\circ$ The point $O$ in the middle of the park is the center of the circle. Now answer the following questions:
$i.$ What is the value of $\angle\text{B}?$
$a. 80^\circ $
$b. 100^\circ $
$c. 90^\circ $
$d. 70^\circ $
$ii.$ What is the value of $\angle\text{C}?$
$a. 80^\circ $
$b. 100^\circ $
$c. 90^\circ $
$d. 70^\circ $
$iii.$ What is the special type of quadrilateral $\text{ABCD}?$
$a.$ Square.
$b.$ Rectangle.
$c.$ Cyclic quadrilateral.
$d.$ Trapezium.
$iv.$ What is the property of cyclic quadrilateral?
$a.$ Opposite angles are supplementary.
$b.$ Adjacent angles are equal.
$c.$ Opposite angles are equal.
$d.$ Adjacent angles are complementary.
$v.$ What you will call the yellow shaded shape $\text{OBC}?$
$a.$ Segment.
$b.$ Arc.
$c.$ Chord.
$d.$ Sector.

Answer

$(i)$	$(b)$	$100^\circ $
$(ii)$	$(a)$	$80^\circ $
$(iii)$	$(c)$	Cyclic quadrilateral.
$(iv)$	$(a)$	Opposite angles are supplementary.
$(v)$	$(d)$	Sector.

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Question 74 Marks

Read the Source/ Text given below and answer any four questions: Four students of class$ IX B$ with names Ajay, Babloo, Charan and Deepak are playing a game in a circular playground. All four students are holding radios with speaker and mic. These radios are connected by a wire of equal length that is $11\ m$ $($for each radio$)$. Ajay Asks a question to Babloo. If Babloo gives the correct answer he gets $10$ points and asks a new question to Charan, If he can not answer then he passes the same question to Charan and gets no points. These conditions apply to all four players. After $10$ rounds who gets maximum points, he becomes the winner.

$i.$ What is the radius of the field?
$a. 7\ m$
$b. 14\ m$
$c. 11\ m$
$d. 22\ m$
$ii.$ What is the area of the field?
$a. 70\ m^2$
$b. 154\ m^2$
$c. 110\ m^2$
$d. 220\ m^2$
$iii.$ What is the area of the part \ marked with 1 on the field?
$a. 50\ m^2$
$b. 154\ m^2$
$c. 76\ m^2$
$d. 38.5\ m^2$
$iv.$ What is the circu\ mference of the field?
$a. 22\ m$
$b. 14\ m$
$c. 44\ m$
$d. 28\ m$
4 What is the direct distance fro\ m Ajay to Charan?
$a. 7\ m$
$b. 28\ m$
$c. 15\ m$
$d. 14\ m$

Answer

$i$	$a$	$7\ m$
$ii$	$b$	$154\ m^2$
$iii$	$d$	$38.5\ m^2$
$iv$	$c$	$44\ m$
$v$	$d$	$14\ m$

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Maths Case study (4 Marks)

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