Case study (4 Marks)

Question 14 Marks

Read the Source/ Text given below and answer any four questions:
Chocolate is in the form of a quadrilateral with sides $6\ cm$ and $10\ cm, 5\ cm$ and$ 5\ cm ($as shown in the figure$)$ is cut into two parts on one of its diagonal by a lady. Part-I is given to her maid and part II is equally divided among a driver and gardener.

Length of $BD:$

$9\ cm$
$8\ cm$
$7\ cm$
$6\ cm$

Area of $\triangle\text{ABC}:$

$24\ cm^2$
$12\ cm^2$
$42\ cm^2$
$21\ cm^2$

The sum of all the angles of a quadrilateral is equal to:

$180^\circ$
$270^\circ$
$360^\circ$
$90^\circ$

A diagonal of a parallelogram divides it into two congruent:

Square.
Parallelogram.
Triangles.
Rectangle.

Each angle of the rectangle is:

More than $90^\circ$
Less than $90^\circ$
Equal to $90^\circ$
Equal to $45^\circ$

Answer

$(i)$	$(b)$	$8\ cm$
$(ii)$	$(a)$	$24\ cm^2$
$(iii)$	$(c)$	$360^\circ $
$(iv)$	$(c)$	Triangles.
$(v)$	$(c)$	Equal to $90^\circ $

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Question 24 Marks

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

Sohan wants to show gratitude towards his teacher by giving her a card made by him. He has three pieces of trapezium pasted one above the other as shown in fig. These pieces are arranged in a way that AB || HC || GD || FE. Also BC = CD = DE and AH = HG = GF = 6cm. He wants to decorate the card by putting up a colored tape on the nonparallel sides of the trapezium.

Find the total length of colored tape required if DE = 4cm.

20cm
30cm
40cm
50cm

ABHC is a trapezium in which AB || HC and $\angle\text{A}=\angle\text{B}=45^\circ.$ Find angles C and H of the trapezium.

135, 130
130, 135
135, 135
130, 130

What is the difference between trapezium and parallelogram?

Trapezium has 2 sides, and parallelogram has 4 sides.
Trapezium has 4 sides, and parallelogram has 2 sides.
Trapezium has 1 pair of parallel sides, and parallelogram has 2 pairs of parallel sides.
Trapezium has 2 pairs of parallel sides, and parallelogram has 1 pair of parallel sides.

Diagonals in isosceles trapezoid are ________.

parallel.
opposite.
vertical.
equal.

ABCD is a trapezium where AB || DC, BD is the diagonal and E is the midpoint of AD. A line is drawn through E parallel to AB intersecting BC at F. Which of these is true?

BF = FC
EA = FB
CF = DE
None of these

Answer

(i)	(b)	30cm
(ii)	(c)	135, 135
(iii)	(c)	Trapezium has 1 pair of parallel sides, and parallelogram has 2 pairs of parallel sides.
(iv)	(d)	equal.
(v)	(a)	BF = FC

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Question 34 Marks

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

There is a Diwali celebration in the DPS school Janakpuri New Delhi. Girls are asked to prepare Rangoli in a triangular shape. They made a rangoli in the shape of triangle ABC. Dimensions of $\triangle\text{ABC}$ are 26cm, 28cm, 25cm.

In fig, R is mid-point of AB and RQ || BC then AQ is equal to:

In fig R and Q are mid-points of AB and AC respectively. The length of RQ is:

14
13
12.5
13.5

If Garland is to be placed along the side of $\triangle\text{QPR}$ which is formed by joining midpoint, what is the length of garland:

79cm
39.5cm
35cm
79.5cm

In the following figure R, P and Q are the mid-points of AB, BC, and AC respectively. Which of the following is the area of $\triangle\text{PQR}?$

$\frac12\text{ar (ABC)}$
$\frac{1}{3}\text{ar (ABC)}$
$\frac14\text{ar (ABC)}$
$\frac16\text{ar (ABC)}$

R, P, Q are the mid-points of corresponding sides AB, BC, CA in $\triangle\text{ABC}$ the figure so obtained BPQR will be:

Parallelogram.
Trapezium.
Quadrilateral.
None of these.

Answer

(i)	(c)	QC
(ii)	(a)	14
(iii)	(b)	39.5cm
(iv)	(c)	$\frac14\text{ar (ABC)}$
(v)	(a)	Parallelogram.

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Question 44 Marks

Read the Source/ Text given below and answer these questions: A class teacher gave students coloured paper in the shape of a quadrilateral. She asks him to make a parallelogram from it using paper folding.

One angle of a quadrilateral is 108° and the remaining three angles are equal, then each of the three equal angles:

90°
74°
84°
72°

How can a parallelogram be formed by using paper folding?

By finding diagonals of the quadrilateral.
By joining mid pts. of sides of a quadrilateral.
By finding angle bisectors.
None of these.

The quadrilateral formed by joining the mid-points of the sides of a quadrilateral PQRS, taken in order, is a rectangle, if:

PQRS is a rectangle.
PQRS is a parallelogram.
diagonals of PQRS are perpendicular.
diagonals of PQRS are equal.

In the figure, ABCD and AEFG are two parallelograms. If $\angle\text{C}=60^\circ,$ then $\angle\text{F}$ is:

30°
60°
90°
120°

Which of the following is not true for a parallelogram?

Opposite sides are equal.
Opposite angles are equal.
Opposite angles are bisected by the diagonals.
Diagonals bisect each other.

The angles of the quadrilateral are in the ratio 2 : 5 : 4 : 1? Which of the following is true?

The largest angle in the quadrilateral is 150°.
The smallest angle is 30°.
The second-largest angle in the quadrilateral is 80°.
Both the largest angle in the quadrilateral is 150° and The smallest angle is 30°.

Answer

(i)	(c)	84°
(ii)	(b)	PQRS is a parallelogram
(iii)	(b)	60°
(iv)	(c)	Opposite angles are bisected by the diagonals.
(v)	(d)	Both the largest angle in the quadrilateral is 150° and The smallest angle is 30°.

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Question 54 Marks

Read the Source/ Text given below and answer any four questions: During Maths Lab Activity each student was given four broomsticks of lengths 8cm, 8cm, 5cm, 5cm to make different types of quadrilaterals.

Using the above information answer the following questions:

How many quadrilaterals can be formed using these sticks?

Only One type of quadrilaterals can be formed.
Two types of quadrilaterals can be formed.
Three types of quadrilaterals can be formed.
Four types of quadrilaterals can be formed.

Name the types of quadrilaterals formed:

Rectangle, parallelogram, kite.
Rectangle, parallelogram, Trapizum.
Rectangle, parallelogram, Square.
Rectangle, Square, kite.

In a trapezium ABCD, DC || AB and $\angle\text{A}=\angle\text{B}=45^\circ,$ the teacher asked the student to find $\angle\text{D}.$ Naresh answered it is ________.

105°
108°
135°
125°

While discussing the properties of a parallelogram teacher asked about the relation between two angles x and y of a parallelogram as shown in fig. The teacher gave them 4 options as (if BC < CD):

x > y
x < y
x = y
None of these.

P, Q, R, and S are respectively the mid-points of sides AB, BC, CD, and DA of quadrilateral ABCD in which AC = BD and $\text{AC}\bot\text{BD}, \text{PQRS},$ is a:

Square.
Rhombus.
Kite.
Parallelogram.

Answer

(i)	(c)	Three types of quadrilaterals can be formed.
(ii)	(a)	Rectangle, parallelogram, kite.
(iii)	(c)	135°
(iv)	(b)	x < y
(v)	(a)	Square.

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Question 64 Marks

In the quadrilateral ABCD given below, ∠DAC = 90° and AB = AC = AD = DE = EB.

8. What is the value of ∠EAC?
A. 15°
B. 30°
C. 45°
D. 90°
9. Which type of quadrilateral is ABCE?
A. Rhombus
B. Kite
C. Trapezium
D. Parallelogram
10. What is the value of ∠ABE?
A. 20°
B. 30°
C. 45°
D. 60°

Answer

8. A. 15°
9. B. Kite
10. B. 30°

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Question 74 Marks

Atul likes to observe the stars with his telescope. He likes to track the movements of stars in the sky.
He took a picture of the night sky one day. On that day, Mars was equidistant from Saturn and Jupiter.

He draws a circle such that the dots showing the planets Mars (M), Jupiter (J), Saturn (S) and a star Altair (A) lies on the boundary of a circle and ∠SMJ = 150°.
1. What is the measure of ∠SAJ?
A. 30°
B. 45°
C. 150°
D. 210°
2. Atul claims that the quadrilateral MJAS is a kite.
What additional information is required to conirm his claim?
A. Distance between Altair and Saturn is equal to the distance between Mars and Jupiter.
B. Distance between Altair and Jupiter is equal to the distance between Mars and Saturn.
C. Distance between Altair and Saturn is equal to the distance between Altair and Mars.
D. Distance between Altair and Saturn is equal to the distance between Altair and Jupiter.
3. The adjacent sides of quadrilateral A are equal to corresponding sides of Quadrilateral B. All angles of Quadrilateral A measure 90°. The angles of Quadrilateral B are 120°, 60°, 120° and 60° respectively.
Which quadrilateral has a greater area? Give reasons.
4. Sanya has a triangular piece of land. She wants to divide it into four equal areas. Suggest a way to do so.
5. Does joining four distinct points always produce a quadrilateral? Justify your answer.

Answer

1. A. 30°
2. D. Distance between Altair and Saturn is equal to the distance between Altair and Jupiter.
3. Mention Quadrilateral A along with a valid mathematical reason.
● Quadrilateral A, both the quadrilaterals have an equal base but the altitude of Quadrilateral A is greater.
4. Accept a valid mathematical division.
● Sanya can ind mid-points of the sides of the triangular region and create a smaller triangular region by connecting them. In this way, the triangular region can be divided into four triangles of equal area.
● Sanya can divide one side into four equal parts and connect each point on the base to the vertex (this may be a more practical way if all the land owners need some part touching the road for access).
5. No, with valid justiication.
No, there can be three cases.
● When all the points are collinear, the resulting igure is a line.
● When three points are collinear out of four, the resulting igure is a triangle.
● When no three points out of four are collinear, the resulting igure is a quadrilateral.

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Question 84 Marks

The figure below shows the side view of a shopping trolley. The metal plate is fixed on the side by the store keeper for advertisement.

6. Three angles of the basket are obtuse. Which type of angle is the fourth?
A. Acute
B. Obtuse
C. Right
D. Relex
7. What is the shape of the metal plate?
A. Square
B. ectangle
C. Rhombus
D. Parallelogram

Answer

6. A. Acute
7. D. Parallelogram

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MATHS Case study (4 Marks)

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