Case study (4 Marks)

Question 14 Marks

Read the Source/ Text given below and answer these questions: In the middle of the city, there was a park 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:

What is the value of $\angle\text{x}?$

110°
70°
90°
100°

$\triangle\text{APD}$ and $\triangle\text{BCQ}$ are congruent by which criteria?

PD is equal to which side?

$\triangle\text{ABC}$ and $\triangle\text{ACD}$ are congruent by which criteria?

What is the value of $\angle\text{m}?$

110°
70°
90°
20°

Answer

(i)	(b)	70°
(ii)	(c)	ASA
(iii)	(d)	BQ
(iv)	(a)	SSS
(v)	(d)	20°

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Question 24 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 5m fell down on the ground. Branch AC makes an angle of 30° with the main tree AP. The distance of Point B from P is 4m. You can observe that $\triangle\text{ABP}$ is congruent to $\triangle\text{ACP}.$

Now answer the following questions:

$\triangle\text{ACP}$ and $\triangle\text{ABP}$ are congruent by which criteria?

What is the length of CP?

What is the value of $\angle\text{BAP}?$

40°
50°
30°
60°

What is the value of $\angle\text{APB}?$

40°
50°
60°
90°

What is the height of the remaining tree?

Answer

i	d	RHS
ii	a	4m
iii	c	30°
iv	d	90°
v	c	3m

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Question 34 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:

Drew a triangle ABC.
D and E are found as the mid points of AB and AC.
DE was joined and DE was extended to F so DE = EF.
FC was joined.

Answer the following questions:

$\triangle\text{ADE}$ and $\triangle\text{EFC}$ are congruent by which criteria?

$\angle\text{EFC}$ is equal to which angle?

$\angle\text{DAE}$
$\angle\text{ADE}$
$\angle\text{AED}$
$\angle\text{B}$

$\angle\text{ECF}$ is equal to which angle?

$\angle\text{DAE}$
$\angle\text{ADE}$
$\angle\text{AED}$
$\angle\text{B}$

CF is equal to which of the following?

CF is parallel to which of the following?

Answer

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

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Question 44 Marks

In a toy game, a robot starts from Home, picks an object from the Shop, delivers it to the Client and goes back Home.

8. Which is the longest segment of the path travelled by the robot? Write the correct words.
9. Rita says, ‘For two triangles to be congruent, any three parameters of the six (3 sides and 3 angles) should be equal.’
Give examples in favour of and against her statement.
10. ‘Two triangles with a pair of equal angles are congruent.’
Why is it necessary to have the side between the two angles be of the same length for both the
triangles?

Answer

8. Shop, Client
Client, Shop
9. Valid mathematical argument including criteria of congruent triangles.
● Example in favour: Side-Side-Side and Angle-Side-Angle criteria
● Example against: Side-Angle-Angle and Angle-Angle-Angle criteria
10. Valid mathematical argument including the construction of triangles and a counter example showing that only angles are not suficient criteria for determining congruence of triangles.
● When speciied angles are drawn at two endpoints of a line segment, they meet at a unique point. If side length and end angles are provided, they will make unique triangles.
● All equilateral triangles are not congruent but have equal angles.

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Question 54 Marks

In the given figure, ∆AFB ≅ ∆AFG, ∆ADE ≅ AGE and ∠EAF = 45°.

1. What is the measure of ∠DAB?
A. 60°
B. 90°
C. 120°
D. 135°
2. What is the length of AD?
3. What is the area of the shaded region?
A. 12.5 cm²
B. 15 cm²
C. 20 cm²
D. 36 cm²

Answer

1. B. 90° °
2. 6
6 cm
3. B. 15 cm²

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Question 64 Marks

The picture below shows a staircase outside a house. Each step of the staircase is congruent and there are 25 steps in the staircase from the loor to the platform and 25 steps from the platform to the roof.

7. What is the length of the staircase railing?

Answer

7. $2 \sqrt{89} m$

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Question 74 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°
B. 60°
C. 70°
D. 80°
5. What is the value of y?
6. What is the value of ∠BDC?
A. 30°
B. 40°
C. 50°
D. 70°

Answer

4. B. 60°
5. 40
40°
6. A. 30°

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Question 84 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 10m 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 10m so that it can reach at point P on the pole and this should make 60° with line AC. Answer the following questions:

In the $\triangle\text{PAC}$ and $\triangle\text{PBC}$ which side is common?

In the $\triangle\text{PAC}$ and $\triangle\text{PBC}$ which angles are given to be equal?

$\angle\text{A}=\angle\text{X}$
$\angle\text{B}=\angle\text{X}$
$\angle\text{B}=\angle\text{Y}$
None

In the figure, $\triangle\text{PAC}$ and $\triangle\text{PBC}$ are congruent due to which criteria?

What is the value of $\angle\text{PBC}\text{?}$

30°
60°
90°
45°

What is the value of $\angle\text{X}\text{?}$

45°
60°
90°
30°

Answer

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

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Maths Case study (4 Marks)

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