Incentre and Its Construction

Q. The point of concurrence of angle bisectors of a triangle is known as ____________.

1. circumcentre
2. incentre
3. origin
4. collinear point

Q.

If $A,B$ and $C$ are interior angles of a triangle $ABC$, then show that $\sin \left(\frac{B+C}{2}\right)=\cos \frac{A}{2}$.

Q. The circumcentre of a triangle lies outside the triangle.

1. Always
2. Never
3. Inside in acute angle triangle, Outside in obtuse angle triangle
4. Can't say

Q.

State whether the statement is True or False.

AAS congruence criterion is same as ASA congruence criterion.

Q. The given figure is formed by 2 quarter circles and the area of the shaded square is 49 cm${}^2$. Find the perimeter of the figure. $(\text{Take}\pi =\frac{22}{7})$

1. 94 cm
2. 108 cm
3. 86 cm
4. 122 cm

Q.

If two lines are transversal to three parallel lines, then the number of points of intersection can be ___ .

Q.

The area of a circular playground is $22176m^2$. Find the cost of fencing this ground at the rate of $Rs50permeter$.

Q.

If the angles of a triangle are in the ratio $1:3:5$ and $\theta$denote the smallest angle, then the ratio of the largest side to the smallest side of the triangle is:

1. $\frac{(\sqrt{3}\sin \theta +\cos \theta )}{2\sin \theta }$

2. $\frac{(\sqrt{3}\cos \theta -\sin \theta )}{2\sin \theta }$

3. $\frac{(\cos \theta +\sqrt{3}\sin \theta )}{2\sin \theta }$

4. $\frac{(\sqrt{3}\cos \theta +\sin \theta )}{2\sin \theta }$

Q. (a) What will be the value of x in the following diagram? [3 MARKS]


(b) If a $△$QPR is right angled at P, which is the longest side of the triangle?

Q. The incentre of a triangle is equidistant from its-

1. Vertices
2. Sides
3. A and B
4. None of these

Q. The incentre of the triangle formed by the axes and the line $\frac{x}{a}+\frac{y}{b}=1$ is

1. $(\frac{a}{2},\frac{b}{2})$
2. $(\frac{ab}{a+b+\sqrt{ab}},\frac{ab}{a+b+\sqrt{ab}})$
3. $(\frac{a}{3},\frac{b}{3})$
4. $(\frac{ab}{a+b+\sqrt{a^2+b^2}},\frac{ab}{a+b+\sqrt{a^2+b^2}})$

Q. Find the value of $x$ in degree. $SQ$ is the angle bisector of $\angle PSR$ and $P$ & $R$ are perpendiculars on side $SP$ and $SR$ from $Q.$

1. ${10}^o$
2. ${11}^o$
3. ${12}^o$
4. ${13}^o$

Q. LN is the angle bisector of ${45}^o$. Find $\angle NLK.$

1. ${22.5}^o$
2. ${23.5}^o$
3. ${24}^o$
4. none of the above

Q. For a right angled triangle ABC(right angled at A), what is the location of the incentre?

1. on A
2. Inside ABC
3. on BC
4. Outside ABC

Q.

Question 38

In the above figure (Q. No. 37), which one of the following is not true?
(a) $\angle 1+\angle 5={180}^{\circ }$
(b) $\angle 2+\angle 5={180}^{\circ }$
(c) $\angle 3+\angle 8={180}^{\circ }$
(d) $\angle 2+\angle 3={180}^{\circ }$

Q. In a $\Delta ABC,$ AD is the bisector of the angle A meeting BC at D. If I is the incentre of the triangle, then AI : DI is equal to -

1. (sinB + sinC) : sinA
2. (cosB + cosC) : cosA
3. $cos\left(\frac{B+C}{2}\right):cos\left(\frac{B+C}{2}\right)$
4. $cos\left(\frac{B-C}{2}\right):cos\left(\frac{B+C}{2}\right)$

Q. What angle do you get if you bisect the angle ${72}^{\circ }$?

1. ${30}^{\circ }$
2. ${42}^{\circ }$
3. ${144}^{\circ }$
4. ${36}^{\circ }$

Q. Find the equation of the bisector of angle $A$ of the triangle whose verities are $A(4,3),B(0,0)$ and $C(2,3)$.

Q.
$\angle CAB={60}^{\circ }$, the value of $\angle CBD$ is

1. 120

Q. Find the coordinates of the incentre of the triangle, equations of whose sides are
$x+1=0,-3x+4y+5=0,5x+12y=27$

Q. In figure, $ABC$ is a right triangle right angled at $B$ such that $BC=6cm$ and $AB=8cm$. Find the radius of its incircle

Q. Where does circumcentre of an obtuse angled triangle lie?

Q. In Fig., AB = AC and $CP\parallel BA$ and AP is the bisector of exterior $\angle CAD$ of $\Delta ABC$. Prove that $ABCP$ is a parallelogram.

Q. $PM$ is the angle bisector of $\angle P$. Find $\angle QPM.$

1. ${45}^o$
2. ${30}^o$
3. ${25}^o$
4. none of these

Q. Find the co-ordinate of te circumcentre of triangle whose vertices are (8, 6), (8-2) and (2, -2). Also its circum radius

Q.

Consider the following figure. The value of $y$ is

1. 60°

2. 45°

3. 90°

4. Data insufficient – the value of x is required

Q. If the vertices of a triangle are $(2,-2),(-1,-1)$ and $(5,2)$, then the equation of its circumcircle is

1. $x^2+y^2+3x+3y+8=0$
2. $x^2+y^2-3x-3y-8=0$
3. $x^2+y^2-3x+3y+8=0$
4. $Noneofthese$

Q. The incentre of the triangle formed by the point $(0,0),(5,12),(16,12)$ is

1. $(6,9)$
2. $(7,9)$
3. $(6,7)$
4. $(9,7)$

Q. The circumcentre of a triangle whose vertices are $A(1,3),B(5,5),C(7,5)$

1. $(2,\frac{-1}{2})$
2. $(\frac{3}{2},\frac{\sqrt{3}-3}{6})$
3. $(6,-2)$
4. $(\frac{5}{4},\frac{-1}{2})$

Q. Show that (2, 1) is the circum-centre of the triangle formed by the vertices (3, 1), (2, 2) and (1, 1).