Home
Class 9
MATHS
A triangle A B C is an isosceles triangl...

A triangle `A B C` is an isosceles triangle if any one of the following conditions hold: Altitude `A D` bisects `/_B A Cdot` Bisector of `/_B A C` is perpendicular to the base `B Cdot`

Text Solution

Verified by Experts

In `triangleADB` and `triangleADC`,
`/_BAD=/_CAD` (`AD` is bisector of `/_BAC`)
`AD=AD` (common)
`/_ADB=/_ADC` (Each equal to `90^@)`
`/_\ADB~=/_\ADC` (By `ASA` congruence criterion)
`AB=AC` (By `CPCT`)
Hence, `/_\ABC` is an isosceles.
Promotional Banner

Topper's Solved these Questions

  • CIRCLE

    RD SHARMA|Exercise All Questions|306 Videos

  • CONSTRUCTION

    RD SHARMA|Exercise All Questions|29 Videos

Similar Questions

Explore conceptually related problems

A triangle ABC is an isosceles triangle if any one of the following conditions hold: Altitude AD bisets /_BAC Bisector of /_BAC is perpendicular to the base BC

A triangle ABC is an isosceles triangle if any one of the following conditions hold: Bisector of /_BAC is perpendicular to the base BC

Prove that ABC is isosceles if any one of the following holds: Altitude AD bisects BC Median AD is perpendicular to the base BC

If A B C is an isosceles triangle such that A B=A C and A D is an altitude from A on B C . Prove that (i) /_B=/_C (ii) A D bisects B C (iii) A D bisects /_A

In Figure, A B C is a triangle in which /_B=2/_C . D is a point on side B C such that A D bisects /_B A C a n d A B=C DdotB E is the bisector of /_B . The measure of /_B A C is 72^0 (b) 73^0 (c) 74^0 (d) 96^0

In any triangle ABC, prove that following: b sin B-C sin C=a sin(B-C)

In any triangle ABC, prove that following: quad b cos B+c cos C=acos(B-C)

A B C\ a n d\ D B C are two isosceles triangles on the same bas B C and vertices A\ a n d\ D are on the same side of B C . If A D is extended to intersect B C at P , show that A P bisects "\ "/_A as well as /_D A P is the perpendicular bisector of B C

Delta ABC is an isosceles triangle in which A B= A C . Side BA is produced to D such that A D = A B (see Fig. 7.34). Show that /_B C D is a right angle.

RD SHARMA-CONGRUENT TRIANGLE -All Questions
  1. Show that the difference of any two sides of a triangle is less than...

    Text Solution

    |

  2. The sum of two angles of a triangle is 80^@ and their difference is 20...

    Text Solution

    |

  3. A triangle A B C is an isosceles triangle if any one of the followin...

    Text Solution

    |

  4. In a A B C , if 2/A=3/B=6/C , determine /A ,/B and /Cdot

    Text Solution

    |

  5. In a right angled triangle, one acute angle is double the other. Pro...

    Text Solution

    |

  6. The sides A B and A C of a A B C are produced to P and Q respectively....

    Text Solution

    |

  7. In Figure, B D and C E are two altitudes of a A B C such that B D=C...

    Text Solution

    |

  8. The side B C\ of a A B C is produced, such that D is one ray B Cdot ...

    Text Solution

    |

  9. In Figure, A P and B Q are perpendiculars to the line segment A B and ...

    Text Solution

    |

  10. In Figure, /X=62^0,/X Y Z=54^0dot If Y O and Z O are bisectors of /X Y...

    Text Solution

    |

  11. AD and BE are respectively altitudes of triangle ABC ...

    Text Solution

    |

  12. In Figure, /Q P R=/P Q R and M and N are respectively on sides Q...

    Text Solution

    |

  13. In Figure, line segment A B is parallel to another line segment C D . ...

    Text Solution

    |

  14. In figure, A D and B E are respectively altitudes of an isosceles tr...

    Text Solution

    |

  15. In figure, P Q R is a triangle in which P Q=P R and S is any point on ...

    Text Solution

    |

  16. If the bisector of the exterior vertical angle of a triangle be par...

    Text Solution

    |

  17. Prove that Two right triangles are congruent if the hypotenuse and o...

    Text Solution

    |

  18. In given figure the altitudes AD, BE and CF , the altitudes of triang...

    Text Solution

    |

  19. A point O is taken inside an equilateral four sided figure A B C D suc...

    Text Solution

    |

  20. In Delta P Q R , if P Q=Q R and L ,M and N are the mid-points ...

    Text Solution

    |