Home
Class 12
MATHS
In a triangle ABC, AD, BE and CF are the...

In a triangle ABC, AD, BE and CF are the altitudes and R is the circum radius, then the radius of the circel DEF is

A

`R/2`

B

2R

C

R

D

none of these

Text Solution

Verified by Experts

Since BE and CF are altitudes of `Delta` ABC. Therefore, the circle described on BC as a diameter passes through E and F and its radius is `a/2`
Thus, the radius of the circumcircle of `Delta FBE` is a/2
Also, `angleFBC=90^(@)-A` `[becauseABE` is right angled at E`]`
`thereforeFE=2(a/2)sin(90^(@)-A)[becausea/(sinA)=b/(sinB)=c/(sinC)=2R]`
`rArrFE=acosA`

Let `R_(1)` be the radius of the circumcircle of `Delta DEF`. Then,
`R_(1)=(FE)/(2sinangleFDE)`
`rArrR_(1)=(acosA)/(2sin(180^(@)-2A))` `[because` Angles of pedal triangleDEF are `180^(@)-2A,180^(@)-2B` and `180^(@)-2C]`
`rArrR_(1)=(acosA)/(2sin2A)=a/(4sinA)=R/2`
Promotional Banner

Topper's Solved these Questions

  • PROPERTIES OF TRIANGLES AND CIRCLES CONNECTED WITH THEM

    OBJECTIVE RD SHARMA ENGLISH|Exercise SOLVED MCQ|2 Videos

  • PROPERTIES OF TRIANGLES AND CIRCLES CONNECTED WITH THEM

    OBJECTIVE RD SHARMA ENGLISH|Exercise Section II - Assertion Reason Type|13 Videos

  • PROPERTIES OF TRIANGLES AND CIRCLES CONNECTED WITH THEM

    OBJECTIVE RD SHARMA ENGLISH|Exercise Chapter Test|55 Videos

  • PLANE AND STRAIGHT LINE IN SPACE

    OBJECTIVE RD SHARMA ENGLISH|Exercise Chapter Test|31 Videos

  • REAL FUNCTIONS

    OBJECTIVE RD SHARMA ENGLISH|Exercise Chapter Test|58 Videos

Similar Questions

Explore conceptually related problems

If in a DeltaABC, AD, BE and CF are the altitudes and R is the circumradius, then the radius of the circumcircle of DeltaDEF is

In a triangle ABC, AB = AC. Show that the altitude AD is median also.

In an acute angle triangle ABC, AD, BE and CF are the altitudes, then (EF)/a+(FD)/b+(DE)/c is equal to -

In a scalene triangle ABC , AD and CF are the altitudes drawn from A and C on the sides BC and AB repectively.If the area of the triangle ABC and BDF are 18sq.units and 2 sq. units respectively and DF = 2sqrt2 , then R =

In an acute angled triangle ABC , let AD, BE and CF be the perpendicular opposite sides of the triangle. The ratio of the product of the side lengths of the triangles DEF and ABC , is equal to

In a right angled triangle ABC, if r(in radius)=7 cm and R(Circumradius) = 32.5 cm, then the area of the triangle (in square cm ) is………………..

In Delta ABC, angle A = (pi)/(3) and its incircle of unit radius. Find the radius of the circle touching the sides AB, AC internally and the incircle of Delta ABC externally is x , then the value of x is

If R be circum-radius of a Delta, then circum-radius of a pedal Delta is

If AD,BE and CF are the altitudes of a triangle ABC whose vertex A is the point (-4,5) . The coordinates of the points E and F are (4,1) and (-1,-4) respectively, then equation of BC is

The radius of the circle passing through the vertices of the triangle ABC, is

OBJECTIVE RD SHARMA ENGLISH-PROPERTIES OF TRIANGLES AND CIRCLES CONNECTED WITH THEM-Section I - Solved Mcqs
  1. In a triangle ABC if r(1) = 36 , r(2) = 18 and r(3) = 12, then the are...

    Text Solution

    |

  2. In a triangle ABC if r(1) = 36 , r(2) = 18 and r(3) = 12 , then the pe...

    Text Solution

    |

  3. In a triangle ABC, AD, BE and CF are the altitudes and R is the circum...

    Text Solution

    |

  4. In a !ABC if a = 7 , b = 8 and c = 9 , then the length of the line joi...

    Text Solution

    |

  5. If the perimeter of a triangle and the diameter of an ex-circle are eq...

    Text Solution

    |

  6. If D id the mid-point of the side BC of a triangle ABC and AD is perpe...

    Text Solution

    |

  7. ABC is a triangle. D is the middle point of BC. If AD is perpendicular...

    Text Solution

    |

  8. If the median of a triangle through A is perpendicular to AB, then

    Text Solution

    |

  9. In a triangle ABC , if r(1) =2r(2) =3r(3) , then a:b:c =

    Text Solution

    |

  10. In Delta ABC, if r(1) lt r(2) lt r(3), then find the order of lengths ...

    Text Solution

    |

  11. In a triangle ABC if r(1) = 8, r(2) = 12 and r(3) = 24, then a =

    Text Solution

    |

  12. If I is the incentre of a DeltaABC such that angleA=60^(@), then AI =

    Text Solution

    |

  13. If I(1) is the centre of the escribed circle touching side BC of !ABC ...

    Text Solution

    |

  14. In a triangle ABC , if 2R + r = r(1) , then

    Text Solution

    |

  15. The sides of a triangle are sin alpha, cos alpha, sqrt(1+sin alpha cos...

    Text Solution

    |

  16. In a triangle ABC , if 3a = b + c , then cot B/2 cot C/2 =

    Text Solution

    |

  17. If sin^(2).(A)/(2), sin^(2).(B)/(2), and sin^(2).(C)/(2) are in H.P., ...

    Text Solution

    |

  18. In a triangle ABC , if 1/(b+c)+1/(c+a)=3/(a+b+c), then angleC=

    Text Solution

    |

  19. Observe the following statements: (I) In !ABC,bcos^(2)C/2+ccos^(2)B/2=...

    Text Solution

    |

  20. In a triangle , if r(1) = 2r(2) = 3r(3), then a/b + b/c + c/a is equa...

    Text Solution

    |