Inradius, Exradii and Circumradius

IMPORTANT

Inradius, Exradii and Circumradius: Overview

This topic covers concepts, such as, Circles Associated with a Triangle, Circum-circle of a Triangle, Circum-radius of a Regular Polygon & In-radius of a Regular Polygon etc.

Important Questions on Inradius, Exradii and Circumradius

EASY
IMPORTANT

The value of 1bc+1ca+1ab is equal to: {where a,b,c denote 3 sides of a ΔABC and r is a in-radius of ΔABC and R is a circum-radius of ΔABC}

MEDIUM
IMPORTANT

In a triangle   ABC,a:b:c=4:5:6. The ratio of the radius of the circumcircle to that of the incircle is: { where a=BC, b=CA, c=AB}

HARD
IMPORTANT

Prove that in any ΔABC,  R+rmina,b,c, where R is the circumradius, r the inradius, and a,b,c the angle bisectors of the triangle.

HARD
IMPORTANT

Divya inscribed a circle inside a regular pentagon, circumscribed a circle around the pentagon, and calculated the area of region between the two circles. Mansi did the same with a regular heptagon. The area of the regions calculated by Divya and Mansi are A and B, respectively. Each polygon had a side length of 2. Which of the following is true?

MEDIUM
IMPORTANT

Let ABC be triangle with AB=AC=6. If the circum radius of the triangle is 5, then BC equals

MEDIUM
IMPORTANT

Let A1A2A3.A9 be a nine-sided regular polygon with side length 2 units. The difference between the lengths of the diagonals A1A5 and A2A4 equals

EASY
IMPORTANT

If R is the circum radius of ΔABC , then AΔABC = ….

EASY
IMPORTANT

The value of 1bc+1ca+1ab is equal to: {where a,b,c denote 3 sides of a ΔABC and r is a in-radius of ΔABC and R is a circum-radius of ΔABC}

MEDIUM
IMPORTANT

In Δ ABC , BC = 13 cm, AC = 14 cm and AB = 15 cm, then its circum-radius is equal to

HARD
IMPORTANT

AD, BE, CF are internal angular bisectors of ABC and I is the incentre. If a(b+c)secA2ID+b(a+c)secB2IE+c(a+b)secC2IF=kabc, then the value of k is:

HARD
IMPORTANT

Let H be the orthocentre of triangle ABC. Then the angle subtended by side BC at the centre of incircle of CHB is:

MEDIUM
IMPORTANT

  ΔABC r is inradius and r1, r2, r3 are exradii opposite to vertex A, vertex B and , vertex C respectively
If r1 = 8, r2 = 12, r3 = 24, then r is equal to 

MEDIUM
IMPORTANT

In a triangle 1-r1r21-r1r3=2, then the triangle is ( r1, r2 and rare exradii of traingle)

HARD
IMPORTANT

In any ABC, the line joining the circumcentre (O) and incentre (I) is parallel to AC, then OI is equal to-

EASY
IMPORTANT

Find the area of the circumcircle of ABC, if b=2, B=30°.

EASY
IMPORTANT

In a triangle, if r1 > r2 > r3.Tick the appropriate option.

MEDIUM
IMPORTANT

Let a, b, and c be the side lengths of a triangle ABC and assume that ab and ac. If x=b+c-a2, then find the minimum value of axrR, where r and R denote inradius and circumradius of triangle ABC.

HARD
IMPORTANT

If 1-r1r21-r1r3=2, then prove that the triangle is right-angled.