Constructions Test

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Constructions Test - 12

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Weekly Quiz Competition

Question 1
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To draw an angle of $$150^{\circ}$$ using a pair of compass and ruler

A
bisect $$120^{\circ}$$ and $$180^{\circ}$$

B
bisect $$60^{\circ}$$ and $$120^{\circ}$$

C
bisect $$0^{\circ}$$ and $$60^{\circ}$$

D
None

Solution
Question 2
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A $$\triangle ABC$$ in which $$AB= 5.4\ \text{cm}, \angle CAB= 45^{\circ}$$ and $$AC + BC= 9\ \text{cm}$$. Then, perimeter of $$\Delta ABC$$ is

A
$$14.4\ \text{cm}$$

B
$$11.4\ \text{cm}$$

C
$$12.4\ \text{cm}$$

D
$$15.4\ \text{cm}$$

Solution

$$\Rightarrow$$ Draw line segment $$AB=5.4\,cm$$
$$\Rightarrow$$ Take $$A$$ as center and draw angle of $$45^o$$
$$\Rightarrow$$ Cut off $$AD=9\,cm$$
$$\Rightarrow$$ Join $$BD$$
$$\Rightarrow$$ Now, draw perpendicular bisector of $$BD$$ which
intersect $$AD$$ at point $$C$$.
$$\Rightarrow$$ Now join $$CB$$
$$\Rightarrow$$ Measure the length of $$AC$$ and $$BC$$
$$\Rightarrow$$ We get $$AC=5\,cm$$ and $$BC=4\,cm$$
$$\Rightarrow$$ Perimeter of $$\triangle ABC=5.4+4+5=14.4\,cm$$
Question 3
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Directions For Questions

Construct a triangle whose perimeter is $$9.8\ cm$$ and the base angles are $$\displaystyle 45^{\circ} $$ and $$ \displaystyle 60^{\circ} $$.

...view full instructions

The value of $$\angle BAC$$ is

A
$$35^\circ$$

B
$$65^\circ$$

C
$$75^\circ$$

D
$$85^\circ$$

Solution

$$\Rightarrow$$ Draw a line segment $$PQ = AB + BC + CA =9.8\,cm.$$
$$\Rightarrow$$ Make $$\angle LPQ= 45^o$$ and $$\angle MQP = 60^o$$
$$\Rightarrow$$ Now bisect $$\angle LPQ$$ and $$\angle MQP$$ such that their bisectors meet at $$A.$$
$$\Rightarrow$$ Draw perpendicular bisector of $$AP$$ and $$AQ$$ i.e.
$$XY$$ and $$ST$$ intersecting $$PQ$$ at $$B$$ and $$C$$ respectively.
$$\Rightarrow$$ Join $$AB$$ and $$AC$$
$$\Rightarrow$$ Now, measure the $$\angle BAC$$
$$\Rightarrow$$ We get, $$\angle BAC=75^o$$.
Question 4
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Directions For Questions

Construct a triangle whose perimeter is $$9.8\ cm$$ and the base angles are $$\displaystyle 45^{\circ} $$ and $$ \displaystyle 60^{\circ} $$.

...view full instructions

Then, length (in $$cm$$) of each of the sides of the triangle is

A
$$3, 4, 4$$

B
$$2.7, 3.7, 3.4$$

C
$$2.3, 4.2, 3.6$$

D
$$4, 1.8, 4$$

Solution

$$\Rightarrow$$ Draw a line segment $$PQ = AB + BC + CA =9.8\,cm.$$
$$\Rightarrow$$ Make $$\angle LPQ= 45^o$$ and $$\angle MQP = 60^o$$
$$\Rightarrow$$ Now bisect $$\angle LPQ$$ and $$\angle MQP$$ such that their bisectors meet at $$A.$$
$$\Rightarrow$$ Draw perpendicular bisector of $$AP$$ and $$AQ$$ i.e.
$$XY$$ and $$ST$$ intersecting $$PQ$$ at $$B$$ and $$C$$ respectively.
$$\Rightarrow$$ Join $$AB$$ and $$AC$$
$$\Rightarrow$$ Now, measure the length of $$\triangle ABC$$
$$\Rightarrow$$ We get, $$AB=3.4\,cm,\,BC=3.7\,cm,\,AC=2.7\,cm$$
Question 5
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Given an angle $$\theta$$, which of the following angles cannot be obtained by using the method of construction of angle bisectors?

A
$$\dfrac{\theta}2$$

B
$$\dfrac{\theta}4$$

C
$$\dfrac{\theta}6$$

D
$$\dfrac{\theta}8$$

Solution

Angle bisector is bisects or divide the angle into two equal parts.
If $$\theta$$ is given, we can find $$\dfrac{\theta}{2}$$ by bisecting $$\theta$$ into two parts and $$\dfrac{\theta}{2}$$ will be half of $$\theta$$.
By construting angle bisector of angle $$\dfrac{\theta}{2}$$, we can get $$\dfrac{\theta}{4}$$ which is half of $$\dfrac{\theta}{2}$$.
And then again bisecting $$\dfrac{\theta}{4}$$, we can get the angle $$\dfrac{\theta}{8}$$ which is half of $$\dfrac{\theta}{4}$$.
Hence, by using method of construction of angle bisectors we can find $$\dfrac { \theta }{ 2 } , \dfrac { \theta }{ 4 } ,\dfrac { \theta }{8} $$ but not $$\dfrac { \theta }{ 6 } $$ as it is half of $$\dfrac{\theta}{3}$$ which can't be determined by angle bisector.
Hence, the answer is $$\dfrac{\theta}{6}$$.
Question 6
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How many angle bisectors need to be drawn in the steps of construction of an angle $$60^\circ$$?

A
$$0$$

B
$$1$$

C
$$2$$

D
$$3$$

Solution
Question 7
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For constructing a triangle whose perimeter and both base angles are given, the first step is to:

A
Draw a base of any length

B
Draw the base of length $$=$$ perimeter

C
Draw the base angles from a random line.

D
Draw a base of length $$=\dfrac13 \times$$ perimeter.

Solution

Below are the steps for constructing a triangle whose perimeter and base angles are given.

Step 1 : Draw a line segment/base equal to perimeter

Step 2 : From any point X draw ray at one of the given base angles. From any point Y draw ray at second of the given base angles

Step 3 : draw angle bisector of X and Y, two angle bisectors intersect each other at point A

Step 4 : Draw line bisector of XA and AY respectively these two line bisectors intersect XY at point B and C

Step 5 : Join A to B and A to C

Step 6 : Triangle ABC is the required triangle.

Hence, the answer is 'Draw the base of length $$=$$ perimeter'.
Question 8
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You are asked to "construct" an angle of $$90^\circ$$. Which of the following methods is considered appropriate for construction

A
Using a compass and straightedge, copy an angle that appears to be close to $$90^\circ$$ from a diagram in your textbook.

B
Using a compass and straightedge, construct two parallel lines and label on of the angles $$90^\circ$$.

C
Using a compass and straightedge, construct angles of $$60^\circ$$ and $$120^\circ$$ and bisect the angle between them.

D
Using a protractor, draw an angle of $$90^\circ$$

Solution

An angle is drawn most appropriate using a compass.
Now bisector of $$60^{\circ}$$ and $$120^{\circ}$$ gives an angle of $$90^{\circ}$$
So option $$C$$ is correct.
It can also be drawn using protractor but less accurate then compass.
Question 9
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How many angle bisectors need to be drawn in the steps of construction of an angle $$45^\circ$$?

A
$$0$$

B
$$1$$

C
$$2$$

D
$$3$$

Solution

Steps for construction:
$$1)$$ Construct a perpendicular line.
$$2)$$ Place a compass on a intersection line.
$$3)$$ Adjust compass width to reach start point
$$4)$$ Draw an arc that intersects perpendicular line.
$$5)$$ Place a ruler on a start point and where arc intersects perpendicular line.
$$6)$$ Draw a 45 degree line
So only $$1$$ angle bisector is need to be drawn in the steps of construction of an angle $$45^o$$
Question 10
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The diagram represents the construction of triangle ABC with which of the following dimensions?

A
$$ \displaystyle BC=4.8 $$ $$cm$$, $$ \displaystyle \angle B=60^{\circ} $$ and $$ \displaystyle AB-AC=12\ cm $$.

B
$$ \displaystyle BC=4.8 $$ $$cm$$, $$ \displaystyle \angle B=30^{\circ} $$ and $$ \displaystyle AB-AC=15\ cm $$.

C
$$ \displaystyle BC=4.8 $$ $$cm$$, $$ \displaystyle \angle B=30^{\circ} $$ and $$ \displaystyle AB-AC=12\ cm $$.

D
None of these

Solution