Tag: construction of tangents

Questions Related to construction of tangents

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

maths construction circumscribing and inscribing a circle on a regular hexagon construction of tangent to a circle construction of tangents construction of line segment and circle of given radius construction related to lines

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

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

  3. $42$

  4. $41$

  5. $40$

Show answer
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

maths construction circumscribing and inscribing a circle on a regular hexagon construction of tangent to a circle construction of tangents

  1. $88^\circ$

  2. $78^\circ$

  3. $68^\circ$

  4. $58^\circ$

Show answer
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.)

maths construction circumscribing and inscribing a circle on a regular hexagon construction of tangent to a circle construction of tangents

  1. $6\ cm$

  2. $7\ cm$

  3. $8\ cm$

  4. $9\ cm$

Show answer
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:

maths construction circumscribing and inscribing a circle on a regular hexagon construction of tangent to a circle construction of tangents construction of line segment and circle of given radius construction related to lines

  1. $1$

  2. $2$

  3. $3$

  4. $4$

Show answer
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 

maths construction circumscribing and inscribing a circle on a regular hexagon construction of tangent to a circle construction of tangents construction of line segment and circle of given radius construction related to lines

  1. $3$

  2. $4$

  3. $2$

  4. $5$

Show answer
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:

maths construction circumscribing and inscribing a circle on a regular hexagon construction of tangent to a circle construction of tangents construction of line segment and circle of given radius construction related to lines

  1. $1$

  2. $2$

  3. $3$

  4. $4$

Show answer
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 

maths construction circumscribing and inscribing a circle on a regular hexagon construction of tangent to a circle construction of tangents construction of line segment and circle of given radius construction related to lines

  1. True

  2. False

Show answer
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

maths construction circumscribing and inscribing a circle on a regular hexagon construction of tangent to a circle construction of tangents construction of line segment and circle of given radius construction related to lines

  1. 3

  2. 4

  3. 2

  4. 5

Show answer
Correct Option: A
Explanation:

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

The sum of all the angles of a pentagon are

maths construction circumscribing and inscribing a circle on a regular hexagon construction of tangent to a circle construction of tangents

  1. $360^\circ$

  2. $540^\circ$

  3. $720^\circ$

  4. none of these

Show answer
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:

maths construction circumscribing and inscribing a circle on a regular hexagon construction of tangent to a circle construction of tangents

  1. $54^\circ$

  2. $60^\circ$

  3. $162^\circ$

  4. $108^\circ$

Show answer
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