0

Construction - class-VIII

Attempted 0/30 Correct 0 Score 0

The number of independent measurement required to construct a triangle is -

  1. $3$

  2. $4$

  3. $6$

  4. $2$


Correct Option: A
Explanation:

Ans- Three measurements are necessarily required to construct a triangle.

 Whether it can be measurements of all three sides,or all three angles or both

The triangle formed by AB = 3 cm BC = 5 cm AC = 9 cm is__

  1. An equilateral triangle

  2. An isosceles triangle

  3. A scalene triangle

  4. None of these


Correct Option: C
Explanation:
Given,  
In $\Delta ABC$  
AB=3cm, BC=5cm and AC=9cm.
Since all sides of triangle ABC are different. So $\Delta$ ABC is a Scalene triangle.

When constructing an inscribed regular hexagon, how will you choose the arc measurement?

  1. radius of the circle

  2. diameter of the circle

  3. chord of the circle

  4. circumference of the circle


Correct Option: A
Explanation:

When constructing an inscribed regular hexagon in a circle, we choose radius of the circle as a arc measurement.

The measure of maximum possible exterior angle in a regular polygon is 

  1. $70^o$

  2. $60^o$

  3. $90^o$

  4. $120^o$


Correct Option: D
Explanation:
Exterior angle of regular polygon = 180-interior angle
exterior angle is maximum when interior angle is minimum.
And we have minimum interior angle for regular triangle that is 60 degree..
So maximum exterior angle will be 180-60=120
So correct answer is Option D

To construct a quadrilateral minimum of its _________ elements are required.

  1. $3$

  2. $4$

  3. $5$

  4. $2$


Correct Option: C
Explanation:

To construct a unique quadrilateral, we will be need a minimum of $5$ dimensions.

If we have five dimensions, we can draw a side first then mark angle on both ends then we can construct a quadrilateral uniquely.
Or if we have three sides and two included angles then also we can construct a unique quadrilateral.
Unless we are constructing any one of the special quadrilaterals.

How many equal parts you will cut the circle to draw inscribing hexagon?

  1. $4$

  2. $5$

  3. $6$

  4. $7$


Correct Option: C
Explanation:

Hexagon is a $6$-sided polygon.
So we will cut the circle into $6$ equal parts.

Which tool will you use for cutting a circle into 6 equal parts?

  1. compass

  2. ruler

  3. protector

  4. divider


Correct Option: A
Explanation:

We will use compass tool for cutting a circle into $6$ equal parts.

While constructing a circle circumscribing and inscribing a regular hexagon, identify the statement true for the construction?

  1. circle outside and inside hexagon

  2. only hexagon is constructed

  3. only circle is drawn

  4. outside hexagon and inside circle


Correct Option: A
Explanation:

While constructing a circle circumscribing and inscribing a regular hexagon, the statement true for the construction is circle outside and inside hexagon.

State true or false:
A quadrilateral is uniquely determined if any four of its elements are known.

  1. True

  2. False


Correct Option: B
Explanation:

Above statement is false.

To construct a unique quadrilateral, we will be need a minimum of 5 dimensions.
If we have five dimensions, we can draw a side first then mark angle on both ends then we can construct a quadrilateral uniquely.

Or if we have three sides and two included angles then also we can construct a unique quadrilateral.
Unless we are constructing any one of the special quadrilaterals.

If the side of a regular hexagon is $6$ cm, then its area will be

  1. $108$ sq. cm

  2. $\dfrac {108}{3}$ sq. cm

  3. $108\sqrt {3}$ sq. cm

  4. $54\sqrt3$ sq. cm


Correct Option: D
Explanation:

$\Rightarrow$  Length of side of regular hexagon $(a)=6\,cm$

$\Rightarrow$  Area of regular  hexagon $=\dfrac{3\sqrt{3}}{2}a^2$

                                                 $=\dfrac{3\sqrt{3}}{2}\times (6)^2$

                                                 $=\dfrac{3\sqrt{3}}{2}\times 36$ 
                                 
                                                 $=54\sqrt{3}\,cm^2$

The centre of the circle circumscribing the square whose three sides are $3x+y=22,x-3y=14$ and $3x=y=62$ is:

  1. $\left( \dfrac { 3 }{ 2 } ,\dfrac { 27 }{ 2 } \right) $

  2. $\left( \dfrac { 27 }{ 2 } ,\dfrac { 3 }{ 2 } \right) $

  3. $(27,3)$

  4. $\left( 1,\dfrac { 2 }{ 3 } \right) $


Correct Option: B

A square is inscribed in the circle $x^2 + y^2 -2x +4y - 93 = 0$ with its sides parallel to the coordinates axes. The coordinates of its vertices are 

  1. $( - 6, - 9), \, ( - 6, 5), \, (8, - 9)$ and $(8, 5)$

  2. $( - 6, 9), \, ( - 6, - 5), \, (8, - 9)$ and $(8, 5)$

  3. $( - 6, - 9), \, ( - 6, 5), \, (8, 9)$ and $(8, 5)$

  4. $( - 6, - 9), \, ( - 6, 5), \, (8, - 9)$ and $(8, - 5)$


Correct Option: A

For each of the following, drawn a circle and inscribe the figure given.If a polygon of the given type can't be inscribed,write not possible.

  1. Rectangle.

  2. Trapezium.

  3. Obtuse triangle.

  4. non-rectangle parallelogram

  5. Accute isosceles triangle.

  6. A quadrilateral PQRS with $\overline {PR} $ as diameter.


Correct Option: A

In regular hexagon, if the radius of circle through vertices is r, then length of the side will be

  1. $\displaystyle \frac{2\pi r}{6}$

  2. r

  3. $\displaystyle \frac{\pi r}{6}$

  4. $\displaystyle \frac{r}{2}$


Correct Option: B
Explanation:

$\Rightarrow$   Radius of a circle is $r$.

$\Rightarrow$   In regular hexagon all sides are equal.
$\Rightarrow$   The regular hexagon has 6 equilateral triangles. The diameter of the circle is $2r$ in this case, will coincide with 2 equilateral triangles. So the side of the hexagon will be $r$.
$\therefore$   Length of side of hexagon is $r$.

When constructing the circles circumscribing and inscribing a regular hexagon with radius $3$ m, then inscribing hexagon length of each side is

  1. $1m$

  2. $2m$

  3. $3m$

  4. $4m$


Correct Option: C
Explanation:

When constructing the circles circumscribing and inscribing a regular hexagon with radius $3$ m, then inscribing hexagon length of each side is $3$ m.

The area of a circle inscribed in a regular hexagon is $100\pi$. The area of the hexagon is:

  1. $600$

  2. $300$

  3. $200\sqrt { 2 } $

  4. $200\sqrt { 3 } $

  5. $200\sqrt { 5 } $


Correct Option: D
Explanation:

Area of circle $=100\pi $
$\pi r^{2}=100\pi $
$r^{2}=100$
$r=10$
Now, a regular hexagon is made up of 6 equilateral $\bigtriangleup s $ of equal areas. Now, height of equilateral $\bigtriangleup  $ is equal to radius of circle.Therefore, ar. of 1 equilateral $\bigtriangleup=\dfrac {1}{2} $ x base x height
$\Rightarrow \dfrac {\sqrt{3}}{4}a^{2}=\dfrac {1}{2}a*10\Rightarrow a=\dfrac {4*10}{2\sqrt{3}}=\dfrac {20\sqrt{3}}{3} $
Area of hexagon $6
\left ( \dfrac {\sqrt{3}}{4}a^{2} \right )=6*\dfrac {\sqrt{3}}{4}\dfrac {20\sqrt{3}}{3}\dfrac {20\sqrt{3}}{3}=200\sqrt{3}$

A circle is inscribed in a quadrilateral ABCD in which $\angle B = 90^o$. If $AD = 23 cm$, $AB = 29 cm$ and $DS = 5 cm$. Find the radius of the circle.

  1. $11$ cm

  2. $13$ cm

  3. $9$ cm

  4. None of these


Correct Option: A
Explanation:

$AS$ and $AP$ are tangents drawn to the circle at $A$

$\implies AS = AP$

Similarly

$BP = BQ$

$QC = CR$

$RD = DS$

Given

$AD = 23$

$\implies AS + SD = 23$

$AS = 23 – 5 = 18 = AP$

$AB = 29 \implies AP + BP = 29$

$\implies 18 + BP = 29 \implies BP = 11cm$

Now consider rectangle $PBQO$

$PB – BQ , OP = OQ = radius$

$\angle PBQ = 90$    

WKT

$OP \perp BP $ and $OQ \perp BQ$

Since radius is perpendicular to tangent at point of contact

$\implies$ All the angles are 90 degree and adjacent sides are equal

So, It is a square

$\implies r = BP = 11cm$

Given are the steps are construction of a pair of tangents to a circle of radius $4$cm from a point on the concentric circle of radius $6$cm. Find which of the following step is wrong?
(P) Take a point O on the plane paper and draw a circle of radius OA$=4$cm. Also, draw a concentric circle of radius OB$=6$cm.
(Q) Find the mid-point A of OB and draw a circle of radius BA$=$AO. Suppose this circle intersects the circle of radius $4$cm at P and Q.
(R) Join BP and BQ to get the desired tangents from a point B on the circle of radius $6$ cm.

  1. Only (P)

  2. Only (Q)

  3. Both (P) & (Q)

  4. Both (Q) & (R)


Correct Option: B

What are the tools required for constructing a tangent to a circle?

  1. ruler

  2. compass

  3. pencil

  4. all the above


Correct Option: D
Explanation:

The tools required for constructing a tangent to a circle is ruler, compass and pencil.

Let C be the circle with centre at $(1, 1)$ and radius $=1$. If T is the circle centred at $(0, y)$, passing through origin and touching the circle C externally, then the radius of T is equal to?

  1. $\dfrac{\sqrt{3}}{\sqrt{2}}$

  2. $\dfrac{\sqrt{3}}{2}$

  3. $\dfrac{1}{2}$

  4. $\dfrac{1}{4}$


Correct Option: A

The sides of a triangle are $25,39$ and $40$. The diameter of the circumscribed circle is: 

  1. $\cfrac { 133 }{ 3 } $

  2. $\cfrac { 125 }{ 3 } $

  3. $42$

  4. $41$

  5. $40$


Correct Option: B
Explanation:

Circum radius formula

$R$ $=\cfrac { abc }{ \sqrt { (a+b+c)(b+c-a)(c+a-b)(a+b-c) }  }$ .
Where  $a, b, c$  are sides of triangle 
$\Rightarrow$ $R$ $=\cfrac { 25\times 39\times 40\quad  }{ \sqrt { (140\quad \times (54)\times (26)\quad \times (240) }  } $
$=\cfrac { 25\times 39\times 40\quad  }{ \sqrt { { 2 }^{ 3 } } \times 13\times 2\times { 3 }^{ 3 }\times 2\times 13\times { 2 }^{ 3 }\times 3 } $
$=\cfrac { 25\times 39\times 40\quad  }{ \sqrt { { 2 }^{ 8 } } \times { 3 }^{ 4 }\times { 13 }^{ 2 } } $.
$=\cfrac { 25 \times \ 39 \times 40  }{ { 2 }^{ 4 }\times { 3 }^{ 2 }\times { 13 } } =\quad \cfrac { 25 \times 39 \times40\quad  }{ 16\times 9\times { 13 } }$ 
$=\cfrac { 125 }{ 6 }$ 
$\therefore$   Diameter $=\cfrac { 125\times \ 2 }{ 6 } = \cfrac { 125 }{ 3 } $

$\therefore$ B) Answer.

The angles of a pentagon in degrees are $y^\circ$, $(y+20^\circ)$, $(y+40^\circ)-(y+60^\circ)$ and $(y+80^\circ)$. The smallest angle of the pentagon is

  1. $88^\circ$

  2. $78^\circ$

  3. $68^\circ$

  4. $58^\circ$


Correct Option: C
Explanation:

Consider the given angles.

${{y}^{\circ }},\left( {{y}^{\circ }}+{{20}^{\circ }} \right),\left( {{y}^{\circ }}+{{40}^{\circ }} \right),\left( {{y}^{\circ }}+{{60}^{\circ }} \right),\left( {{y}^{\circ }}+{{80}^{\circ }} \right)$

 

We know that the sum of all angles of pentagon

$ {{y}^{\circ }}+\left( {{y}^{\circ }}+{{20}^{\circ }} \right)+\left( {{y}^{\circ }}+{{40}^{\circ }} \right)+\left( {{y}^{\circ }}+{{60}^{\circ }} \right)+\left( {{y}^{\circ }}+{{80}^{\circ }} \right)={{540}^{\circ }} $

$ 5{{y}^{\circ }}+{{200}^{\circ }}={{540}^{\circ }} $

$ 5{{y}^{\circ }}={{340}^{\circ }} $

 

Hence, the smallest angle of the pentagon is ${{68}^{\circ }}$.

Construct a regular pentagon inside a circle of radius $6\ cm$. The length of each side of the pentagon is: (approx.)

  1. $6\ cm$

  2. $7\ cm$

  3. $8\ cm$

  4. $9\ cm$


Correct Option: B
Explanation:

Each side of the pentagon makes an angle x at the center

$\implies 5x= 360 $

$x = 72$

Now lets consider side AB which is a chord to the circle

Let OP be a perpendicular to AB

$\implies AP = BP \implies AB = 2AP$

IN $\triangle OAP$

$\angle OPA = 90$

$\angle POA = \dfrac{x}{2} = \dfrac{72}{2} = 36$

$\sin 36 = \dfrac{AP}{OA}$

$AP = 0.6 \times 6 = 3.6$

$AB = 2 \times 3.6 = 7cm$

The minimum number of dimensions needed to construct an equilateral triangle is:

  1. $1$

  2. $2$

  3. $3$

  4. $4$


Correct Option: A
Explanation:

As we know that all angles in an equilateral triangle measures $60^o$. Hence we need only the length of the side to construct an equilateral triangle.

The number of independent measurement required to construct a $\Delta$ le is 

  1. $3$

  2. $4$

  3. $2$

  4. $5$


Correct Option: A
Explanation:

Triangle has $3$ sides.
So, number of measurements required to construct a triangle is $3$.

The minimum number of dimensions needed to construct a rectangle is:

  1. $1$

  2. $2$

  3. $3$

  4. $4$


Correct Option: B
Explanation:

We can construct a rectangle when:

(i) two adjacent sides are given
(ii) one side and the diagonal is given
(iii) both diagonals are given
In the above cases the number of dimensions needed to construct a rectangle is $2$.

State True or False
There is a triangle whose sides have lengths 10.2 cm, 5.8 cm and 4.5 cm 

  1. True

  2. False


Correct Option: A
Explanation:

Suppose such a triangle is possible Then the sum of the lengths of any two side would be greater than the length of the third side  Let us check this
Is 4.5+5.8>10.2  Yes 
Is 5.8+10.2>4.5  Yes
Is 10.2+4.5>5.8  Yes
Therefore the triangle is possible

The number of independent measurements required to construct a $\Delta$ is

  1. 3

  2. 4

  3. 2

  4. 5


Correct Option: A
Explanation:

We have three measurements to construct a $\Delta$ le,

The sum of all the angles of a pentagon are

  1. $360^\circ$

  2. $540^\circ$

  3. $720^\circ$

  4. none of these


Correct Option: B
Explanation:

Pentagon is a five sided polygon.

The sum of the interior angles of the pentagon is the sum of interior angles of the three triangles.The sum of interior angles of the three triangles is 180 degree.so the sum of interior angles of the pentagon is 3 times 180 degree which is 540 degree.

Inscribe a regular pentagon in a circle of radius $3\ cm$. The interior angles of the pentagon are:

  1. $54^\circ$

  2. $60^\circ$

  3. $162^\circ$

  4. $108^\circ$


Correct Option: D
Explanation:

We know that internal angle of regular pentagon is $\cfrac{(n-2)}{n}180^{\circ}$ where n = number of sides.

Here, n = 5.
So, interior angle is $\cfrac{(5-2)}{5}180^{\circ} = 108^{\circ}$

So correct answer is option D

- Hide questions