Construction of Incircle

Q.

Evaluate$\cos (\mathrm{\pi }-\mathrm{x})$

Q. A circle touches all the four sides of a quadrilateral ABCD. Prove that AB + CD = BC + DA

Q. Prove that the area of an equilateral triangle described on one side of the square is equal to half the area of the equilateral triangle described on one of its diagonal.
OR
if the area of two similar triangles are equal, prove that they are congruent.

Q.

From a point P which is at a distance of 13 cm from the centre O of a circle of radius 5 cm, the pair of tangents PQ and PR to the circle is drawn. Then, the area of the quadrilateral PQOR is
(A) 60 cm${}^2$
(B) 65 cm${}^2$
(C) 30 cm${}^2$
(D) 32.5 cm${}^2$

Q. Question 2
Let s denotes the semi – perimeter of a ∆ ABC, in which BC=a, CA=b and AB=c , if a circle touches the sides BC, CA , AB at D, E, F respectively prove that BD = s - b.

Q.

Which of these triangles have the circumcentre and incentre at the same point?

1. Scalene Triangle

2. Equilateral Triangle

3. Isosceles Triangle

4. Obtuse Triangle

Q.

If O is a point inside a triangle ABC, which of the following is true ?

1. AO + BO + CO > (AB + BC + CA)

2. AO + BO + CO = AB + BC + CA

3. None of these

4. (AO + BO + CO) < (AB + BC + CA)

Q. Question 8
If a number of circles touch a given line segment PQ at a point A, then their centres lie on the perpendicular bisector of PQ.

Q. In an equilateral triangle, the point of intersection of all angle bisectors is:

1. Incentre
2. Orthocentre
3. Circumcentre
4. All of the above

Q. Prove that the lengths of tangents drawn from an external point to the circle are equal.

Q.

For a triangle ABC, which of the following is not true?

1. Difference of the two sides of the triangle is equal to the third side

2. Difference of any two sides of a triangle is always less than the third side.

3. If a line parallel to one side of a triangle intersects the other two sides in different points, it cuts off segments proportional to the sides.

4. Two triangles similar to a third triangle are similar to each other.

Q.

Prove that the perimeter of a right triangle is equal to the sum of the diameter of its incircle and twice the diameter of its circumcircle.

Q. In a right angled triangle PQR, Circumcenter and orthocenter are at a distance of ------ units from each other. (Where R is the circumradius)

1. R

2. 2R

3. 3R

4. 4R

Q. In a right triangle ABC, $\angle$ACB = 90$°$ a circle is inscribed in the triangle with radius r.
a, b, c are the lengths of the side BC, AC and AB respectively.Prove that 2r = a+b$-$c

Q.

The distance between the tops of two trees 20 m and 28 m high is 17 m. The horizontal distance between the trees is

1. 11 m

2. 31 m

3. 15 m

4. 9 m

Q. Question 6
In a circular table cover of radius 32 cm, a design is formed leaving an equilateral triangle ABC in the middle as shown in Fig. 12.24. Find the area of the design.

Q. The sides of a triangle are $6cm,11cm$ and $15cm$. The radius of its incircle is:

1. $\frac{5\sqrt{2}}{4}cm$
2. $3\sqrt{2}cm$
3. $6\sqrt{2}cm$
4. $\frac{4\sqrt{2}}{5}cm$

Q.
AP is the tangent to the circle with centre O and radius 8 cm. If AB = 9 cm, then the length of the tangent AP is .

1. 10 cm
2. 13 cm
3. 15 cm

Q. O is the centre of the incircle for $△$ABC. Find $\angle$BOC if $\angle$BAC = 40${}^o$.

Q. The ratio of the area of the circumcircle and incircle of an equilateral triangle is:

1. $4:1$
2. $1:4$
3. $9:1$
4. $1:9$

Q.

In triangle $ABC,a^2+c^2=2002b^2$, then $\frac{cotA+cotC}{cotB}$ is equal to

1. $\frac{1}{2001}$

2. $\frac{2}{2001}$

3. $\frac{3}{2001}$

4. $\frac{4}{2001}$

Q.

$\overleftrightarrow{CX}$ is a tangent to circle with centre A, at a point X. $\overline{AX}$ = 5 cm, CX = 12 cm, AC = ?

1. 12 cm

2. 4 cm

3. 9 cm

4. 13 cm

Q. If Δ ABC is isosceles with AB = AC and C (O, r) is the incircle of the ΔABC touching BC at L, prove that L bisects BC.

Q.

In figure, a square OABC is inscribed in a quadrant OPBQ. If OA = 20 cm, find the area of the shaded region. (Use π = 3.14)

1. 200

2. 230

3. 245

4. 228

Q. If one vertex of an equilateral triangle of side a is origin and the other lies on the line $x-\sqrt{3}y=0$ then the coordinates of third vertex are

1. (0, a)
2. (0, -a)
3. $[\frac{\sqrt{3}a}{2},-\frac{a}{2}]$
4. $[-\frac{\sqrt{3}a}{2},\frac{a}{2}]$

Q. $ABC$ is a right angle triangle right angled at $A.$ A circle is inscribed in it. The lengths of two sides containing the right angle are $6$ cm and $8$ cm. Find the radius of the circle$?$

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

Q. In the given figure, $△ABC$ is right-angles at $B$, in which $AB=15cm$ and $BC=8cm$. A circle with centre $O$ has been inscribed in $△ABC$. The value of $x$, the radius of the inscribed circle is

1. 6cm
2. 4cm
3. 3cm
4. 9cm

Q. If twice the square of the diameter of a circle is equal to half the sum of the squares of the sides of inscribed triangle $ABC$, then ${\sin }^{2}A+{\sin }^{2}B+{\sin }^{2}C$ is equal to

1. $1$
2. $2$
3. $4$
4. $8$

Q.

In $△$ ABC, if cot A, cot B, cot C be in A. P. then $a^2,b^2,c^2$ are in

1. G.P.

2. H.P.

3. A.P.

4. None of these

Q. Question 7
In figure , BC is a diameter of the circle, AC = 6cm and AB = 8 cm. Find the area of the shaded region. (use $\pi =3.14$)