Case study (4 Marks)

Question 14 Marks

Vijay is trying to find the average height of a tower near his house. He is using the properties of similar triangles. The height of Vijay's house is 20 meter. Wen Vijay's house casts a shadow 10 m long on the ground at the same time, the tower casts a shadows 50 m long on the ground and the house of Ajay casts 20 m shadow on the ground. Based on the above information answer the following questions.

(i) The height of the tower is
(a) 20 m $\qquad$ (b) 50 m $\qquad$ (c) 100 m $\qquad$ (d) 200 m
(ii) When Vijay's house casts a shadow of 12 m , the length of the shadow of the tower is
(a) 75 m $\qquad$ (b) 50 m $\qquad$ (c) 45 m $\qquad$ (d) 60 m
(iii) The height of Ajay's house is
(a) 30 m $\qquad$ (b) 40 m $\qquad$ (c) 50 m $\qquad$ (d) 20 m
(iv) When the tower casts a shadow of 40 m , the length of the shadow of Ajay's house is
(a) 16 m $\qquad$ (b) 32 m $\qquad$ (c) 20 m $\qquad$ (d) 8 m

Answer

Three triangles formed by Vijay's house, tower and Ajay's house are shown in Fig. these are similar triangles.

$\begin{array}{ll}\therefore & \frac{A C}{A B}=\frac{P R}{P Q}=\frac{L N}{L M} \qquad\ldots(i)\\ \Rightarrow & \frac{20}{10}=\frac{P R}{50}=\frac{L N}{20} \Rightarrow 2=\frac{P R}{50} \text { and } 2=\frac{L N}{20} \Rightarrow P R=100 \text { and } L N=40\qquad\ldots(ii)\end{array}$
(i) (c): From (ii), we obtain $P R=$ Height of tower $=100 m$.
(ii) (c): From (i), we obtain
$
\frac{A C}{A B}=\frac{P R}{P Q}
$
$\Rightarrow \quad \frac{20}{12}=\frac{100}{P Q} \qquad[$ From (i), we obtain $P R=100, A C=20$ and $A B=12$ (given)]
$\Rightarrow \quad P Q=60 m$
(iii) (b) : From (ii), we obtain $L N=40 m$.
(iv) (c) : From (i), we obtain $\frac{P R}{P Q}=\frac{L N}{L M}$. When $P Q=40, L N=40$ and $P R=100$, we obtain
$\frac{100}{40}=\frac{40}{L M} \Rightarrow L M=16$

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Question 24 Marks

Observe the below given figures carefully and answer the questions:

(i) Which among the above shown figures are congruent figures?
(a) A and C $\qquad$ (b) $E$ and $F$ $\qquad$ (c) $D$ and $F$ $\qquad$ (d) $B$ and $F$
(ii) Which of the following statements is correct?
(a) All similar figures are congruent.
(b) All congruent figures are similar.
(c) The criterion for similarity and congruency is same.
(d) Similar figures have same size and shape.
(iii) If a line divides any two sides of the triangle in the same ratio, then the line is parallel to the third side. Which theorem is depicted by this statement?
(a) Pythagoras
(b) Thales Theorem
(c) Converse of Thales theorem
(d) Converse of Pythagoras theorem
(iv) Using the concept of similarity, the height of the tree is

(a) 12 ft $\qquad$ (b) 10 ft $\qquad$ (c) 15 ft $\qquad$ (d) 7 ft

Answer

(i) (d) (ii) (b) (iii) (c) (iv) (b)

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Question 34 Marks

In a room a bulb is fixed at a point $O$ on the ceiling. Just below the bulb a large table is placed as shown in Fig. 7.40. A cardboard is cut in the form of quadrilateral $A B C D$ and is fixed between the bulb and the table. When bulb is switched on, shadow $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ of cardboard $A B C D$ is formed on the top of the table such that quadrilateral $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ is an enlargement of quadrilateral $A B C D$ with scale factor $1: 2$. If $A B=1.5 cm, B C=2.5 cm, C D=2.4 cm, A D=2.1 cm$, $\angle A=105^{\circ}, \angle B=100^{\circ}, \angle C=70^{\circ}$ and $\angle D=85^{\circ} cm$, answer the following questions:

(i) The measurement of $\angle A^{\prime}$ is
(a) $105^{\circ}$ $\qquad$ (b) $100^{\circ}$ $\qquad$ (c) $70^{\circ}$ $\qquad$ (d) $80^{\circ}$
(ii) The sum of the angles $\angle A^{\prime}$ and $\angle C^{\prime}$ of quadrilateral $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ is(a) $185^{\circ}$ $\qquad$ (b) $205^{\circ}$ $\qquad$ (c) $175^{\circ}$ $\qquad$ (d) $155^{\circ}$
(iii) Perimeter of quadrilateral $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ is
(a) 8.5 cm $\qquad$ (b) 5 cm $\qquad$ (c) 10 cm $\qquad$ (d) 17 cm
(iv) The length of side $A^{\prime} B^{\prime}$ of quadrilateral $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ is
(a) 1.5 cm $\qquad$ (b) 3 cm $\qquad$ (c) 2.5 cm $\qquad$ (d) 5 cm

Answer

(i) (a) (ii) (c) (iii) (d) (iv) (b)

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Question 44 Marks

Rahul is studying in $X$ standard. He is making a kite to fly it on a Sunday. Few questions came to his mind while making the kite. Give answers to his questions by looking at the figure. (Fig. 7.39).

(i) Rahul tied the sticks at what angles to each other?
(a) $30^{\circ}$ $\qquad$ (b) $60^{\circ}$ $\qquad$ (c) $90^{\circ}$ $\qquad$ (d) $60^{\circ}$
(ii) Which is the correct similarity criteria applicable for smaller triangles at the upper part of this kite?
(a) RHS $\qquad$ (b) SAS $\qquad$ (c) SSA $\qquad$ (d) AAS
(iii) Sides of two similar triangles are in the ratio 4:9. Corresponding medians of these triangles are in the ratio
(a) $2: 3$ $\qquad$ (b) $4: 9$ $\qquad$ (c) $81: 16$ $\qquad$ (d) $16: 81$
(iv) In a triangle, if square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle. This theorem is called as,
(a) Pythagoras theorem
(b) Thales theorem
(c) Converse of Thales theorem
(d) Converse of Pythagoras theorem

Answer

(i) (c) (ii) (b) (iii) (b) (iv) (d)

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Question 54 Marks

A scale drawing of an object is the same shape as the object but a different size. The scale of a drawing is a comparison of the length used on a drawing to the length it represents. The scale is written as a ratio.
Scale factor $=\frac{\text { Length in image }}{\text { Corresponding length in object }}$
If one shape can become another using resizing then the shapes are similar. The ratio of two corresponding sides in similar figures is called the scale factor.

Hence, two shapes are similar when one can become the other after a resize, flip, slide or turn

(i) A model of a boat is made on the scale of $1: 4$. The model is 120 cm long. The full size of the boat has a width of 60 cm . What is the width of the scale model?

(a) 20 cm $\qquad$ (b) 25 cm $\qquad$ (c) 15 cm $\qquad$ (d) 240 cm
(ii) What will effect the similarity of any two polygons?
(a) They are flipped horizontally $\qquad$ (b) They are dilated by a scale factor
(c) They are translated down $\qquad$ (d) They are not the mirror image of one another
(iii) If two similar triangles have a scale factor of $a: b$. Which statement regarding the two triangles is true?
(a) The ratio of their perimeters is $3 a: b$ (b) Their altitudes have a ratio $a: b$
(c) Their medians have a ratio $\frac{a}{2}: b$
(d) Their angle bisectors have a ratio $a^2: b^2$
(iv) The shadow of a stick 5 m long is 2 m . At the same time the shadow of a tree 12.5 m high is
(a) 3 m $\qquad$ (b) 3.5 m $\qquad$ (c) 45 m $\qquad$ (d) 5 m

Answer

(i) (c) (ii) (d) (iii) (b) (iv) (d)

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Maths Case study (4 Marks)

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