Home
Class 9
MATHS
A B C and A D C are two right triangles ...

`A B C` and `A D C` are two right triangles with common hypotenuse `A Cdot` Prove that `/_C A D=/_C B Ddot`

Text Solution

Verified by Experts

Proof: Since `angleABC` and `angleADC` are `90^@`.
These angles are in the semi-circle
Thus, both the triangles are lying in the semi-circle and AC is the diameter of the circle.
Points A,B,C and D are concyclic.
Hence, CD is the chord.
`angleCAD=angleCBD` (Angles in the same segment of the circle)
Promotional Banner

Topper's Solved these Questions

  • AREA OF PARALLELOGRAMS AND TRIANGLES

    RD SHARMA|Exercise All Questions|205 Videos

  • CONGRUENT TRIANGLE

    RD SHARMA|Exercise All Questions|291 Videos

Similar Questions

Explore conceptually related problems

In Figure, A N and C P are perpendicular to the diagonal B D of a parallelogram A B C Ddot Prove that : A D N~= C B P (ii) A N=C P

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 B D\ ~= A C D (ii) A B P\ ~= A C P

In right triangle A B C , right angle at C , M is the mid-point of the hypotenuse A Bdot C is jointed to M and produced to a point D such that D M=C Ddot Point D is joined to point Bdot Show that A M C~= B M D (ii) /_D B C=/_A C B D B C~= A C B (iii) C M=1/2A B

A B C is a triangle. D is a point on A B such that A D=1/4A B and E is a point on A C such that A E=1/4A Cdot Prove that D E=1/4B Cdot

A B C D is a trapezium with A B D Cdot A line parallel to A C intersects A B at X and B C at Ydot Prove that a r( A D X)=a r( A C Y)dot

Triangles A B C and +DBC are on the same base B C with A, D on opposite side of line B C , such that a r(_|_ A B C)=a r( D B C)dot Show that B C bisects A Ddot

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 A B C ,/_B=2/_C and the bisector of /_B intersects A C at Ddot Prove that (B D)/(D A)=(B C)/(B A)

A B C is a triangle in which D is the mid-point of B C and E is the mid-point of A Ddot Prove that area of B E D=1/4a r e aof A B Cdot GIVEN : A A B C ,D is the mid-point of B C and E is the mid-point of the median A Ddot TO PROVE : a r( B E D)=1/4a r( A B C)dot

A B C is a triangle. D is the mid point of B Cdot If A D is perpendicular to A C , then prove that cos A\ dotcos C=(2(c^2-a^2))/(3a c) .

RD SHARMA-CIRCLE -All Questions
  1. Two circle intersect in Aa n dBa n dA Ca n dA D are respectively the d...

    Text Solution

    |

  2. Two circles are drawn with sides A B ,A C of a triangle A B C as diame...

    Text Solution

    |

  3. A B C and A D C are two right triangles with common hypotenuse A Cdot ...

    Text Solution

    |

  4. In the figure, P is the centre of the circle. Prove that: /X P Z=2\...

    Text Solution

    |

  5. In a circle with centre O , chords A B\ a n d\ C D intersect ins...

    Text Solution

    |

  6. C is a point on the minor arc AB of the circle, with centre ...

    Text Solution

    |

  7. In Figure, chord E D is parallel to the diameter A C of the circ...

    Text Solution

    |

  8. In Figure, A B\ a n d\ C D are two chords of a circle, intersecting...

    Text Solution

    |

  9. If O is the circumcentre of a A B C and O D|B C , prove that /B O ...

    Text Solution

    |

  10. In the given figure, a diameter PQ of a circle bisects the chord RS at...

    Text Solution

    |

  11. In Figure, A B=C Ddot Prove that B E=D E\ a n d\ A E=C E , where E is ...

    Text Solution

    |

  12. Two diameters of a circle intersect each other at right angles. Prove ...

    Text Solution

    |

  13. Prove that the circle drawn with any side of a rhombus as a diameter, ...

    Text Solution

    |

  14. A Ca n dB D are chords of a circle that bisect each other. Prove that:...

    Text Solution

    |

  15. A Ca n dB D are chords of a circle that bisect each other. Prove that:...

    Text Solution

    |

  16. A B C and A D C are two right triangles with common hypotenuse A Cdot ...

    Text Solution

    |

  17. D is a point on the circumcircle of A B C in which A B=A C such tha...

    Text Solution

    |

  18. The bisector of /B of an isosceles triangle A B C with A B=A C mee...

    Text Solution

    |

  19. Bisectors of angles A, B and C of a triangle ABC intersect its circum...

    Text Solution

    |

  20. Prove that the mid-point of the hypotenuse of a right triangle is eq...

    Text Solution

    |