Incircle and Its Construction

Q.

In the given figure, $△ABC$ is right-angled at B, in which AB = 15 cm and BC = 8 cm. A circle with center O has been inscribed in $△ABC$. Calculate the value of x, the radius of the inscribed circle. [4 MARKS]

Q.

Find the area of the segment of a circle of radius 12 cm whose corresponding sector has a central angle of ${60}^{\circ }$. (use $\pi =3.14$)

Q.

In the given figure ΔABC is an isosceles traingle with AB=AC and $\angle$ABC = ${50}^{\circ }$. Find $\angle$ BEC = \({x}^{0}\, find the value of x.

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Q.

Find the value of x from the given figure, in which O is the centre of the circle.

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Q.

Draw the incircle of equilateral triangle of side 5 cm. Measure the radius of the circle.

1. 2 cm

2. 3 cm

3. 5 cm

4. 1.4 cm

Q.

O is the centre of the circle. $\angle$OAB = 20${}^{\circ }$, $\angle$OCB = 55${}^{\circ }$. Then$\angle$BOC= and \(\angle)AOC=
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Q.

Congruent polygons have the same area.

1. False

2. True

Q.

Which of the following statements is/are correct?

1. The locus of the point of intersection of two perpendicular tangents is called director circle of the given circle.

2. The director circle of a circle is concentric circle.

3. The radius of director circle of a circle is equal to the radius of original circle.C

1. Only 1 & 2

2. Only 1

3. Only 1 & 3

4. All 1, 2 and 3

Q. Construct a right triangle ABC with AB = 6 cm, BC = 8 cm and $\angle$ B = ${90}^{\circ }$. Draw BD, the perpendicular from B on AC. Draw the circle through B, C and D and construct the tangents from A to this circle. [4 MARKS]

Q.

Prove that the quadrilateral formed by angle bisectors of a cyclic quadrilateral ABCD is also cyclic. [4 MARKS]

Q. 46. Prove that the parallelogram circumscribing a circle is rhombus

Q.

Construct a tangent to a circle of radius 4 cm from a point on the concentric circle of radius 6 cm and measure its length. Also verify the measurement by actual calculation.

Q. 36. Prove that when a parallelogram is circumscribed about a circle, it is a rhombus

Q.

Construct a $\Delta$PQRwith QR = 8 cm, PQ = 6 cm, PR = 4 cm. Construct the incircle of the triangle and find the radius of the incircle.

1. 4 cm

2. 2.2 cm

3. 1.3 cm

4. 5.4 cm

Q.

In the given figure, AB is the diameter of the circle. AC is equal to BC. Find the value of $\angle ABC$.

1. 30°

2. 45°

3. 60°

4. 90°

Q. Match the two column;
$\begin{array}{cc}Column-I& Column-II\\ \left(A\right)Inthegivenfigure,& \left(p\right){60}^{\circ }\\ ABCDisacyclicQuadrilateral.& \\ Oisthecentreofthecircle.& \\ If\angle BOD={160}^{\circ }findtheMeasureof\angle BPD.& \end{array}$


$\begin{array}{cc}\left(B\right)Ingivenfigure,ABCDisacyclic& \left(q\right){65}^{\circ }\\ quadrilateralwhosesideABisadiameter& \\ ofthecirclethroughA,B,C,D.& \\ If\angle ADC={130}^{\circ },find\angle BAC.& \end{array}$


$\begin{array}{cc}\left(C\right)InthegivenfigureBD=DCand& \left(r\right){40}^{\circ }\\ \angle CBD={30}^{\circ }Findm\left(\angle BAC\right)& \end{array}$


$\begin{array}{cc}\left(D\right)Inthegivenfigure,Oisthecentreofthe& \left(s\right){100}^{\circ }\\ arcABCsubtendsanangleof{130}^{\circ }atthecentre.& \\ IfABisextendedtoPmfind\angle PBC.& \end{array}$


Choose the correct option:

1. A-s; B-r; C-p; D-q
2. A-s; B-p; C-q; D-r
3. A-p; B-q; C-r; D-s
4. A-s; B-p; C-r; D-q

Q. The perpendicular drawn from the point of intersection of angle bisectors to any side of a polygon is called

1. circumradius
2. inradius
3. median

Q.

Show that the line segment joining the points of contact of two parallel tangents passes through the centre.

Q. Draw the incircle of equilateral triangle of side 5 cm. Measure the radius of the circle. [3 Marks]

Q. $PQR$ is a right-angled triangle with $PQ=3cm$ and $QR=4cm$. A circle which touches all the sides of the triangle is inscribed in the triangle. The radius of the circle is

1. 2cm
2. 1cm
3. 5cm
4. 3cm

Q. Assertion :If the centroid of an equilateral triangle is $(2,2)$ and its one vertex is $(-3,4)$, then the equation of its circumcircle is $x^2+y^2-4x-4y-21=0$ Reason: Circumcircle coincides with the centroid of an equilateral triangle.

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Both Assertion and Reason are incorrect

Q. Define the incircle of a triangle? How can you locate the incenter of a triangle?
[2 Marks]

Q. If all the sides of a parallelogram touches a circle, then the parallelogram is necessarily a

1. rhombus.
2. square
3. None of the above.
4. rectangle

Q. $ABCD$ is a cyclic quadrilateral. Find the angle A of the cyclic quadrilateral

1. $100$
2. $80$
3. $120$
4. $110$

Q. Match the point of intersection of the given lines.

1. Incentre
2. Orthocentre
3. Centroid

Q.

Use the following information to answer the next question.

How many diagonals containing point B or point D can be drawn in the given polygon?

1. 12

2. 7

3. 9

4. 10

Q. In the given figure, O is a point in the interior of a triangle.

ABC, OD $\perp$ BC, OE $\perp$ AC and OF $\perp$ AB. Then,
$A{F}^2+B{D}^2+C{E}^2=A{E}^2+C{D}^2+B{F}^2$

1. $True$
2. $False$

Q. In the figure below, the rectangle at the corner measures $10cm\times 20cm.$ The corner A of the rectangle is also a point on the circumference of the circle. What is the radius of the circle in cm?

1. 10 cm
2. 40 cm
3. 50 cm
4. 30 cm

Q. The triangle $PQR$ is inscribed in the circle $x^2+y^2=25$. If $Q=(3,4)$ and $R=(-4,3)$ then $\angle QPR=$

1. $\frac{\pi }{4}$
2. $\frac{\pi }{6}$
3. $\frac{3\pi }{4}$
4. $\frac{\pi }{3}$

Q. In the given figure, if $\angle AOB={125}^{\circ }$, then $\angle COD$ is equal to :

1. ${62.5}^{\circ }$
2. ${125}^{\circ }$
3. ${55}^{\circ }$
4. ${45}^{\circ }$