Case study (4 Marks)

Question 14 Marks

Read the following text carefully and answer the questions that follow:
Rohan draws a circle of radius $10 \ cm$ with the help of a compass and scale. He also draws two chords, $AB$ and $CD$ in such a way that the perpendicular distance from the center to $AB$ and $CD$ are $6 \ cm$ and $8 \ cm$ respectively. Now, he has some doubts that are given below.

$i.$ Show that the perpendicular drawn from the Centre of a circle to a chord bisects the chord.
$ii.$ What is the length of $CD?$
$iii.$ What is the length of $AB?$
OR
How many circles can be drawn from given three noncollinear points?

Answer

$i.$ In $\ce{\Delta AOP}$ and $\ce{\Delta BOP}$
$\ce{\angle APO =\angle BPO}($ Given $)$
$\text{OP = OP} ($Common $)$
$\ce{AO = OB} ($ radius of circle $)$
$\ce{\Delta AOP \cong \Delta BOP}$
$\ce{AP = BP (CPCT)}$
$ii.$ In right $\ce{\Delta COQ}$
$\ce{CO^2=OQ^2 + C Q^2}$
$\Rightarrow 10^2=8^2+CQ^2$
$\Rightarrow CQ^2=100-64=36$
$\Rightarrow CQ=6$
$\ce{CD=2CQ}$
$\Rightarrow CD=12 \ cm$
$iii.$ In right $\Delta A O B$
$\ce{AO^2 = OP^2 + AP^2}$
$\Rightarrow 10^2=6^2+A P^2$
$\Rightarrow AP^2=100-36=64$
$\Rightarrow AP =8$
$\ce{AB =2 AP}$
$\Rightarrow AB =16 \ cm$
OR
There is one and only one circle passing through three given non$-$collinear points.

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Question 24 Marks

Read the following text carefully and answer the questions that follow:
A children's park is in the shape of isosceles triangle said $\text{PQR}$ with $\text{PQ = PR, S}$ and $T$ are points on $\text{QR}$ such that $\text{QT = RS}$.

$i.$ Which rule is applied to prove that congruency of $\ce{\triangle PQS}$ and $\ce{\triangle PRT}$.
$ii.$ Name the type of $\ce{\triangle PST.}$
$iii.$ If $\ce{PQ=6 \ cm}$ and $\ce{QR=7 \ cm}$, then find perimeter of $\ce{\triangle PQR}$.
OR
If $\ce{\angle QPR =80^{\circ}}$ find $\ce{\angle PQR}$ ?

Answer

$i.$ In $\ce{\triangle PQS}$ and $\ce{\triangle PRT}$
$\text{PQ = PR}($ Given $)$
$\text{QS = TR}($ Given $)$
$\ce{\angle PQR =\angle PRQ} ($corresponding angles of an isosceles $\triangle)$
By $\text{SAS}$ commence
$\ce{\triangle PQS \cong \triangle PRT}$
$ii.\ce{ \triangle PQS \cong \triangle PRT}$
$\ce{\Rightarrow PS = PT (CPCT)}$
So in $\ce{\triangle PST}$
$\text{PS = PT}$
It is an isosceles triangle.
$iii.$ Perimeter $=$ sum of all $3$ sides
$\text{PQ=PR}=6 \ cm$
$\text{QR}=7 \ cm$
So ,$P =(6+6+7) \ cm$
$=19 \ cm$
OR
Let $\angle Q =\angle R = x$ and $\angle P =80^{\circ}$
In $\ce{\triangle PQR , \angle P +\angle Q +\angle R =180^{\circ}}($ Angle sum property of $\triangle)$
$80^{\circ}+ x + x =180^{\circ}$
$2 x =180^{\circ}-80$
$2 x =100^{\circ}$
$x =\frac{100^{\circ}}{2}$
$=50^{\circ}$

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Question 34 Marks

Read the following text carefully and answer the questions that follow:
Peter, Kevin James, Reeta and Veena were students of Class $9^{th}\ B$ at Govt Sr Sec School, Sector $5,$ Gurgaon.
Once the teacher told Peter to think a number $x$ and to Kevin to think another number $y$ so that the difference of the numbers is $10 (x > y).$
Now the teacher asked James to add double of Peter's number and that three times of Kevin's number, the total was found $120.$
Reeta just entered in the class, she did not know any number.
The teacher said Reeta to form the $1^{st}$ equation with two variables $x$ and $y.$
Now Veena just entered the class so the teacher told her to form $2^{nd}$ equation with two variables $x$ and $y.$
Now teacher Told Reeta to find the values of $x$ and $y.$ Peter and kelvin were told to verify the numbers $x$ and $y.$

$i.$ What are the equation formed by Reeta and Veena?
$ii.$ What was the equation formed by Veena?
$iii.$ Which number did Peter think?
OR
Which number did Kelvin think?

Answer

$i. x-y=10$
$2 x+3 y=120$
$ii. 2 x+3 y=120$
$iii. x-y=10 \ldots(1)$
$2 x+3 y=120 \ldots(2)$
Multiply equation $(1)$ by $3$ and to equation $(2)$
$3 x -3 y +2 x +3 y =30+120$
$\Rightarrow 5 x =150$
$\Rightarrow x =30$
Hence the number thought by Prateek is $30.$
OR
We know that $x - y = 10 ...(i)$ and $2x + 3y = 120 ...(ii)$
Put $x = 30$ in equation $(i)$
$30 - y = 10$
$\Rightarrow y=40$
Hence number thought by Kevin $= 40.$

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Maths Case study (4 Marks)

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