Constructions Test

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Constructions Test - 12

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Weekly Quiz Competition

Question 1
1 / -0

To construct a triangle similar to a given ABC with its sides $$\cfrac{3}{7}$$ of the corresponding sides of $$\Delta$$ ABC, first draw a ray BX such that $$\angle$$CBX is an acute angle and X lies on the opposite side of A with respect to BC. Then locate points $$B_1, B_2, B_3,$$ ... on BX at equal distances and next step is to join

A
$$B_{10}$$to C

B
$$B_{3}to$$ C

C
$$B_{7}$$to C

D
$$B_{4}$$ to C

Solution

To construct the similar $$\Delta$$ with sides $$\cfrac { 3 }{ 7 }$$ of ABC we should divide $$BC$$ in the ratio $$3:7$$.
$$\therefore BX$$ need to have $$7$$ equidistant points on it. because 7 is greater number.
So, the next step is to join $${ B }_{ 7 }$$ to $$C$$
Question 2
1 / -0

To draw a pair of tangents to a circle which are inclined to each other at an angle $$\displaystyle { 35 }^{ \circ }$$, it is required to draw tangents at the end points of those two radii of the circle, the angle between which is

A
$$\displaystyle { 105 }^{ \circ }$$

B
$$\displaystyle { 70 }^{ \circ }$$

C
$$\displaystyle { 145 }^{ \circ }$$

D
$$\displaystyle { 150 }^{ \circ }$$

Solution

Clearly from the figure, by joining radii to both the point of contact, we get quadrilateral & the sum of opposite anlges of a quadrilateral is $$180°$$.
Since, $$\theta = 35^\circ$$
$$\therefore$$ Angle between radii $$=\displaystyle { 180 }^{ \circ }-{ 35 }^{ \circ }={ 145 }^{ \circ }$$
Question 3
1 / -0

Draw a right triangle ABC in which $$BC=12\text{ cm}$$ cm, $$AB= 5\text{ cm}$$ cm and $$\angle B = 90^\circ.$$ Construct a triangle similar to it and of scale factor $$\dfrac{2}{3}$$. Is the new triangle also a right triangle?

A
Yes

B
No

C
Can't say

D
Data insufficient
Solution
In $$ \triangle ABC$$,
$$AB=5\text{ cm}$$, $$BC=12\text{ cm}$$, $$ \angle B={ 90 }^{ \circ }$$

To construct-
A $$ \triangle$$ of scale factor $$\dfrac { 2 }{ 3 } $$ which is similar to $$\triangle ABC$$ as well.

Construction:-
$$ BC$$ is divided into ratio $$2:3$$ at $$ D$$.
$$DE$$ drawn parallel to $$AB$$.
It intersects $$AC$$ at $$E$$.
$$ \triangle DEC$$ is the desired $$\triangle$$.
From the figure we can easily conclude that $$\triangle DEC$$ is a right angle triangle as well.
Question 4
1 / -0

To construct a triangle similar to a given $$\triangle ABC$$ with its sides $${8}/{5}$$ of the corresponding sides of $$\triangle ABC$$ draw a ray $$BX$$ such that $$\angle CBX$$ is an acute angle and $$X$$ is on the opposite side of $$A$$ with respect to $$BC$$. The minimum number of points to be located at equal distances on ray $$BX$$ is

A
5

B
8

C
13

D
3

Solution

1) Draw a triangle ABC.
2) Draw a line BX making an angle with BC.
We will mark 8 points on the Ray BX and points are B1,B2.B3,B4 ..B8.
Question 5
1 / -0

To construct a triangle similar to a given $$\triangle$$ ABC with its sides $$\dfrac{8}{5}$$ of the corresponding sides of $$\triangle ABC$$, draw a ray $$BX$$, such that $$\angle CBX$$ is an acute angle and $$X$$ is on the opposite side of $$A$$ with respect to $$BC$$. The minimum number of points to be located at equal distances on ray $$BX$$ is:

A
5

B
8

C
13

D
3

Solution

To construct a triangle similar to a triangle, with its sides $$\dfrac{m}{n}$$ of the corresponding sides of given triangle, the minimum number of points to be located at an equal distance is equal to $$m$$ or $$n$$, whichever is greater.
Here, $$m : n = 8 : 5$$
or $$ \dfrac{m}{n} = \dfrac{8}{5} $$
So, the minimum number of points to be located at equal distance on Ray $$BX$$ is $$8$$.
Option B is the correct answer.
Question 6
1 / -0

Draw a parallelogram $$ABCD$$ in which $$BC$$ $$=$$ $$5$$ cm, $$AB$$ $$=$$ $$3$$ cm and $$\angle ABC =60^{0}$$. divide it into triangles $$BCD$$ and $$ABD$$ by the diagonal $$BD$$. Construct the triangle $$BD' C'$$ similar to $$BDC$$ with scale factor $$\dfrac{4}{3}$$. Draw the line segment $$D'A'$$ parallel to $$DA$$, where $$A'$$ lies on extended side $$BA$$. Is $$A'BC'D'$$ a parallelogram?

A
Yes

B
No

C
Data insufficient

D
Ambiguous
Question 7
1 / -0

In the construction of triangle similar and larger to a given triangle as per given scale factor m : n, the construction is possible only when :

A
m > n

B
m = n

C
m < n

D
independent of scale factor

Solution

To construct a lager triangle we need $$m>n$$
So that $$\dfrac{m}{n}>1$$ and we get a large scale.
So option $$A$$ is correct.
Question 8
1 / -0

Directions For Questions

$$P$$ is outside circle with center $$O$$.
Here are the steps of construction arranged randomly to construct a pair of tangents from?
$$(a)$$ Taking midpoint of $$OP$$ as center, we draw a circle of radius $$\dfrac{OP}2$$
$$(b)$$ Join $$OP$$
$$(c)$$ Join $$P$$ to the points at which the drawn circle touches the circle with center $$O$$
$$(d)$$ Bisects $$OP$$ and get the midpoint.

...view full instructions

Which is $$2^{nd}$$ step?

A
$$(a)$$

B
$$(b)$$

C
$$(c)$$

D
$$(d)$$

Solution

The correct order of the steps of construction will be:
$$(b)$$ Join $$OP$$
$$(d)$$ Bisect $$OP$$ and get the midpoint.
$$(a)$$ Taking midpoint of $$OP$$ as center, we draw a circle of radius $$\dfrac{OP}2$$
$$(c)$$ Join $$P$$ to the points at which the drawn circle touches the circle with center $$O$$.
Question 9
1 / -0

To construct a triangle similar to given $$\triangle ABC$$ with its sides $$\dfrac23$$ of that of $$\displaystyle \Delta ABC$$, locate points on ray $$BX$$ at equal distances as $$\displaystyle { B }_{ 1 },{ B }_{ 2 },{ B }_{ 3 },....$$ such that $$\displaystyle \angle CBX$$ is acute. The points to be joined in the next step are:

A
$$\displaystyle { B }_{ 4 },C$$

B
$$\displaystyle { B }_{ 3 },C$$

C
$$\displaystyle { B }_{ 1 },C$$

D
$$\displaystyle { B }_{ 2 },C$$

Solution

$$\triangle PQB$$ is the required triangle.
Since side $$BQ$$ is $$\dfrac23$$ times side $$BC$$.
$$BQ = \dfrac23 \times (BQ+CQ) \Rightarrow 3BQ= 2BQ+2CQ \Rightarrow BQ = 2CQ \Rightarrow \dfrac{CQ}{BQ} = \dfrac12$$
Therefore, $$Q$$ divides $$BC$$ in ratio $$2:1$$.
So, point $$B_3$$ should be connected to $$C$$, and $$B_2Q$$ should be drawn parallel to $$B_3C$$.
Question 10
1 / -0

To construct a triangle similar to given $$\displaystyle \Delta ABC$$ with its sides $$\dfrac35$$ of that of $$\displaystyle \Delta ABC$$, a ray $$BX$$ is drawn at acute angle with $$BC$$. How many minimum no. of points should be marked on $$BX$$?

A
$$2$$

B
$$3$$

C
$$4$$

D
$$5$$

Solution

Suppose $$\triangle PQB$$ is the required triangle.
$$BQ = \dfrac35 \times BC\Rightarrow PQ = \dfrac35\times (BQ+QC) \Rightarrow 5BQ = 3BQ+3CQ\Rightarrow 2BQ = 3CQ$$, i.e, $$\dfrac{BQ}{CQ} = \dfrac32$$
Therefore, $$Q$$ divides $$BC$$ in ratio $$3:2$$. So, minimum number of points required on ray $$BX = 2+3=5$$