Home
Class 12
MATHS
In ΔABC, then show that r(r1+r2+r3)=ab+b...

In `ΔABC`, then show that `r(r_1+r_2+r_3)=ab+bc+ac−s^2`.

Text Solution

Verified by Experts

The correct Answer is:
D
In a triangle orthocenter A centriod B and circumcentre C are always collinear such that `AB:BC=2:1`

So, slope of AD is infinity.
Let orthocenter have coordinates (2,k).
Slope of AC is `-2`.
`therefore Slope of BH,(k-2)/(2-5)=(1)/(2)`
`therefore k=(1)/(2)`
`therefore AC=sqrt((6-(-3))^2+(2-5)^2)=sqrt(81+9)=sqrt90=3sqrt10`
AC is diameter of the circle.
`therefore`"Radius of circle" `=(3)/(2)sqrt10=3sqrt(5/(2))`
Promotional Banner

Topper's Solved these Questions

  • COORDINATE SYSYEM

    CENGAGE|Exercise Exercise (Numerical)|12 Videos

  • COORDINATE SYSTEM

    CENGAGE|Exercise Multiple Correct Answers Type|2 Videos

  • CROSS PRODUCTS

    CENGAGE|Exercise DPP 2.2|13 Videos

Similar Questions

Explore conceptually related problems

Prove that r_1+r_2+r_3-r=4R

If I is the incenter of Delta ABC and R_(1), R_(2), and R_(3) are, respectively, the radii of the circumcircle of the triangle IBC, ICA, and IAB, then prove that R_(1) R_(2) R_(3) = 2r R^(2)

In DeltaABC, R, r, r_(1), r_(2), r_(3) denote the circumradius, inradius, the exradii opposite to the vertices A,B, C respectively. Given that r_(1) :r_(2): r_(3) = 1: 2 : 3 The value of R : r is

Prove that (r_1+r_2)/(1+cosC)=2R

In DeltaABC, R, r, r_(1), r_(2), r_(3) denote the circumradius, inradius, the exradii opposite to the vertices A,B, C respectively. Given that r_(1) :r_(2): r_(3) = 1: 2 : 3 The sides of the triangle are in the ratio

In DeltaABC, R, r, r_(1), r_(2), r_(3) denote the circumradius, inradius, the exradii opposite to the vertices A,B, C respectively. Given that r_(1) :r_(2): r_(3) = 1: 2 : 3 The greatest angle of the triangle is given by

In any triangle ABC, find the least value of (r_(1) + r_(2) + r_(3))/(r)

If p+q=1, then show that sum_(r=0)^n r^2^n C_rp^r q^(n-r)=n p q+n^2p^2dot

A,B,C are events such that P_(r )(A)=0.3, P_(r )(B)=0.4,P_(r )(C )=0.8 P_(r ) (AB)=0.08, P_(r )(AC )=0.28 and P_(r )(B)=0.4, P_(r )(C )=0.8 If P_(r )(A cup B cup C) ge 0.75 , then show that P_(r )(BC ) lies in the interval [0.23,0.48].

If in Delta ABC, (a -b) (s-c) = (b -c) (s-a) , prove that r_(1), r_(2), r_(3) are in A.P.