Case study (4 Marks)

Question 14 Marks

In the given figure, the isosceles triangle $ABC ≅ EAD.$ The point $E$ is equidistant from both $A$ and $B.$

$4.$ What is the value of $x?$
$A. 40^\circ $
$B. 60^\circ $
$C. 70^\circ $
$D. 80^\circ $
$5.$ What is the value of $y?$
$6.$ What is the value of $\angle BDC?$
$A. 30^\circ $
$B. 40^\circ $
$C. 50^\circ $
$D. 70^\circ $

Answer

$4. B. 60^\circ $
$5. 40$
$40^\circ $
$6. A. 30^\circ $

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Question 24 Marks

In the given figure, $\triangle AFB ≅ \triangle AFG, \triangle ADE ≅ AGE$ and $\angle EAF = 45^\circ .$

$1.$ What is the measure of $\angle DAB?$
$A. 60^\circ $
$B. 90^\circ $
$C. 120^\circ $
$D. 135^\circ $
$2.$ What is the length of $AD?$
$3.$ What is the area of the shaded region$?$
$A. 12.5\ cm^2$
$B. 15\ cm^2$
$C. 20\ cm^2$
$D. 36\ cm^2$

Answer

$1. B. 90^\circ $
$2. 6$
$6 cm$
$3. B. 15\ cm^2$

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Question 34 Marks

Read the Source/ Text given below and answer any four questions: In a forest, a big tree got broken due to heavy rain and wind. Due to this rain the big branches $AB$ and $AC$ with lengths $5\ m$ fell down on the ground. Branch $AC$ makes an angle of $30^\circ $ with the main tree $AP.$ The distance of Point $B$ from $P$ is $4\ m.$ You can observe that $\triangle\text{ABP}$ is congruent to $\triangle\text{ACP}.$

Now answer the following questions:
$i. \triangle\text{ACP}$ and $\triangle\text{ABP}$ are congruent by which criteria?
$a. \text{SSS}$
$b. \text{SAS}$
$c.\text{ASA}$
$d. \text{RHS}$
$ii.$ What is the length of $CP?$
$a.4\ m$
$b.5\ m$
$c.3\ m$
$d.10\ m$
$iii.$ What is the value of $\angle\text{BAP}?$
$a.40^\circ $
$b.50^\circ $
$c.30^\circ $
$d.60^\circ $
$iv.$ What is the value of $\angle\text{APB}?$
$a.40^\circ $
$b.50^\circ $
$c.60^\circ $
$d.90^\circ $
$v.$ What is the height of the remaining tree$?$
$a.4\ m$
$5\ m$
$c.3\ m$
$d.10\ m$

Answer

$i$	$d$	$RHS$
$ii$	$a$	$4m$
$iii$	$c$	$30^\circ $
$iv$	$d$	$90^\circ $
$v$	$c$	$3m$

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Question 44 Marks

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

Hareesh and Deep were trying to prove a theorem. For this they did the following:
$i.$ Drew a $\triangle ABC.$
$ii. D$ and $E$ are found as the mid points of $AB$ and $AC.$
$iii. DE$ was joined and $DE$ was extended to $F$ so $DE = EF.$
$iv. FC$ was joined.
Answer the following questions:
$i. \triangle\text{ADE}$ and $\triangle\text{EFC}$ are congruent by which criteria$?$
$\text{SSS}$
$\text{RHS}$
$\text{SAS}$
$\text{ASA}$
$ii. \angle\text{EFC}$ is equal to which angle$?$
$a. \angle\text{DAE}$
$b. \angle\text{ADE}$
$c. \angle\text{AED}$
$d. \angle\text{B}$
$iii. \angle\text{ECF}$ is equal to which angle$?$
$a. \angle\text{DAE}$
$b. \angle\text{ADE}$
$c. \angle\text{AED}$
$d. \angle\text{B}$
$iv. CF$ is equal to which of the following$?$
$a. BD$
$b. CE$
$c. AE$
$d. EF$
$v. CF$ is parallel to which of the following$?$
$a. AE$
$b. CE$
$c. BD$
$d. EF$

Answer

$(i)$	$(c)$	$\text{SAS}$
$(ii)$	$(b)$	$\angle\text{ADE}$
$(iii)$	$(a)$	$\angle\text{DAE}$
$(iv)$	$(a)$	$BD$
$(v)$	$(c)$	$BD$

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Question 54 Marks

Read the Source/ Text given below and answer any four questions:

As shown In the village of Surya there was a big pole $PC.$ This pole was tied with a strong wire of $10\ m$ length. Once there was a big spark on this pole, thus wires got damaged very badly. Any small fault was usually repaired with the help of a rope which normal board electricians were carrying on bicycles. This time electricians need a staircase of $10\ m$ so that it can reach at point $P$ on the pole and this should make $60^\circ $ with line $AC.$ Answer the following questions:
$i.$ In the $\triangle\text{PAC}$ and $\triangle\text{PBC}$ which side is common?
$a. PC$
$b. AB$
$c.  AC$
$d. BC$
$ii.$ In the $\triangle\text{PAC}$ and $\triangle\text{PBC}$ which angles are given to be equal?
$a. \angle\text{A}=\angle\text{X}$
$b. \angle\text{B}=\angle\text{X}$
$c.  \angle\text{B}=\angle\text{Y}$
d. None
$iii.$ In the figure, $\triangle\text{PAC}$ and $\triangle\text{PBC}$ are congruent due to which criteria$?$
$a. \text{RHS}$
$b. \text{SAS}$
$c.  \text{SSS}$
$d. \text{ASA}$
$iv.$ What is the value of $\angle\text{PBC}\text{?}$
$a. 30^\circ $
$b. 60^\circ $
$c.  90^\circ $
$d. 45^\circ $
$v.$ What is the value of $\angle\text{X}\text{?}$
$a. 45^\circ $
$b. 60^\circ $
$c.  90^\circ $
$d. 30^\circ $

Answer

$i$	$a$	$PC$
$ii$	$b$	$\angle\text{B}=\angle\text{X}$
$iii$	$a$	$\text{RHS}$
$iv$	$b$	$60^\circ $
$v$	$d$	$30^\circ $

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Question 64 Marks

Read the Source/ Text given below and answer these questions: In the middle of the city, there was a park $\text{ABCD}$ in the form of a parallelogram form so that $AB = CD, AB \| CD$ and $AD = BC, AD \| BC$ Municipality converted this park into a rectangular form by adding land in the form of $\triangle\text{APD}$ and $\triangle\text{BCQ}.$ Both the triangular shape of land were covered by planting flower plants.

Answer the following questions:
$i.$ What is the value of $\angle\text{x}?$
$a. 110^\circ $
$b. 70^\circ $
$c. 90^\circ $
$d. 100^\circ $
$ii. \triangle\text{APD}$ and $\triangle\text{BCQ}$ are congruent by which criteria?
$a. \text{SSS}$
$b. \text{SAS}$
$c. \text{ASA}$
$d. \text{RHS}$
$iii.PD$ is equal to which side$?$
$a. DC$
$b. AB$
$c. BC$
$d. BQ$
$iv. \triangle\text{ABC}$ and $\triangle\text{ACD}$ are congruent by which criteria?
$a. \text{SSS}$
$b. \text{SAS}$
$c. \text{ASA}$
$d. \text{RHS}$
$v.$ What is the value of $\angle\text{m}?$
$a. 110^\circ $
$b. 70^\circ $
$c. 90^\circ $
$d. 20^\circ $

Answer

$(i)$	$(b)$	$70^\circ $
$(ii)$	$(c)$	$ASA$
$(iii)$	$(d)$	$BQ$
$(iv)$	$(a)$	$\text{SSS}$
$(v)$	$(d)$	$20^\circ $

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Maths Case study (4 Marks)

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