Home
Class 9
MATHS
A D ,\ B E\ a n d\ C F , the altitudes o...

`A D ,\ B E\ a n d\ C F ,` the altitudes of ` A B C` are equal. Prove that ` A B C` is an equilateral triangle.

Text Solution

AI Generated Solution

To prove that triangle ABC is an equilateral triangle given that the altitudes AD, BE, and CF are equal, we will follow these steps: ### Step 1: Establish the Given Information We know that AD, BE, and CF are the altitudes from vertices A, B, and C to the opposite sides BC, AC, and AB respectively, and that these altitudes are equal: \[ AD = BE = CF \] ### Step 2: Consider Triangles BEC and BFC We will analyze triangles BEC and BFC. ...
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

In Figure, B D and C E are two altitudes of a A B C such that B D=C Edot Prove that A B C is isosceles. Figure

In Figure A B C D\ a n d\ A E F D are two parallelograms. Prove that: P E=F Q

In Figure, A D\ a n d\ B E are medians of A B C\ a n d\ B E||D F . Prove that C F=1/4A C

Fill in the blanks in the following so that each of the following statements is true Sides opposite to equal angles of a triangle are ....... Angle opposite to equal sides of a triangle are ....... In an opposite to equal sides of a triangle are ....... In a A B C if /_A=/_C , then A B= ..... If altitudes C E a n d B F of a triangle A B C are equal, then A B= ........ In an isosceles triangle A B C with A B=A C , if B D a n d C E are its altitudes, then B D is .......... C E In right triangles A B C a n d D E F , if hypotenuse A B=E F and side A C=D E , then A B C ~= ....

The sides A B\ a n d\ C D of a parallelogram A B C D are bisected at E\ a n d\ Fdot Prove that E B F D is a parallelogram.

In Figure, A B C D ,\ A B F E\ a n d\ C D E F are parallelograms. Prove that a r( A D E)=a r( B C F)

In Figure, it is given that A B=B C and A D=E C . Prove that A B E~=C B D

In Fig. 4.101, A D and C E are two altitudes of A B C . Prove that A E F ~ C D F (ii) A B D ~ C B E (iii) A E F ~ A D B (iv) F D C ~ B E C (FIGURE)

In a A B C ,\ E\ a n d\ F are the mid-points of A C\ a n d\ A B respectively. The altitude A P\ to\ B C intersects F E\ at Qdot Prove that A Q=Q P

A B C is a triangle in which D is the mid-point of B C ,\ E\ a n d\ F are mid-points of D C\ a n d\ A E respectively. If area of A B C is 16\ c m^2, find the area of D E F

RD SHARMA-CONGRUENT TRIANGLE -All Questions
  1. In Figure, it is given that A B=C D\ a n d\ A D=B Cdot Prove that A...

    Text Solution

    |

  2. In A P Q R , if P Q=P Q\ a n d\ L ,\ M\ a n d\ N are the mid-points o...

    Text Solution

    |

  3. A D ,\ B E\ a n d\ C F , the altitudes of A B C are equal. Prove that...

    Text Solution

    |

  4. In Figure, it is given that L M=M N ,\ Q M=M R ,\ M L\ |P Q\ a n d\ ...

    Text Solution

    |

  5. If A B C is an isosceles triangle such that A B=A C and A D is an alti...

    Text Solution

    |

  6. P is a point equidistant from two lines l and m intersecting at poin...

    Text Solution

    |

  7. A B C is a triangle and D is the mid-point of B C . The perpendiculars...

    Text Solution

    |

  8. A B C is a triangle is which B E and C F are, respectively, the perpen...

    Text Solution

    |

  9. In perpendiculars from any point within an angle on its arms are con...

    Text Solution

    |

  10. In Figure, A D\ |C D\ and C B\ |c ddot If A Q=B P\ a n d\ D P=C Q , p...

    Text Solution

    |

  11. A B C D is a square, X\ a n d\ Y are points on sides A D\ a n d\ B C r...

    Text Solution

    |

  12. Which of the following statements are true (T) and which are false (...

    Text Solution

    |

  13. Fill the blanks in the following so that each of the following stateme...

    Text Solution

    |

  14. In a A B C ,\ if /A=45^0\ a n d\ /B=70^0dot Determine the shortest ...

    Text Solution

    |

  15. In a A B C , if /A=50^0\ a n d\ /B=60^0 , determine the shortest a...

    Text Solution

    |

  16. In Figure, P Q > P RdotQ S\ a n d\ R S are the bisectors of L Q and /R...

    Text Solution

    |

  17. In Figure, sides L M\ a n d\ L N OF LMN are extended to P\ a n d\ Q r...

    Text Solution

    |

  18. In Figure, P Q=P Rdot Show that P S > P Q

    Text Solution

    |

  19. In Figure, A B > A Cdot Show that A B > A D

    Text Solution

    |

  20. If D is any point on the base B C produced, of an isosceles tria...

    Text Solution

    |