Home
Class 12
MATHS
In triangleABC, P,Q, R are the feet of a...

In `triangleABC, P,Q, R` are the feet of angle bisectors from the vertices to their opposite sides as shown in the figure. `trianglePQR` is constructed

If `AB = 7` units, BC = 8 units, AC = 5 units, then the side PQ will be

A

`(sqrt28)/(3)` units

B

`(sqrt88)/(3)` units

C

`(sqrt78)/(3)` units

D

`(sqrt84)/(3)` units

Text Solution

Verified by Experts

The correct Answer is:
D
`cos C = ((a^(2) + b^(2) -c^(2))/(2ab)) = (1)/(2)`
In `DeltaPQC` we have
`cos C = (CQ^(2) + PC^(2) - PQ^(2))/(2PC.CQ)`
`rArr PQ^(2) = PC^(2) + CQ^(2) - PC.CQ`
`{"where ", PC = (ab)/(b+c) and CQ = (ab)/(a+c)}`
so `PQ = sqrt((100)/(9) + (64)/(9) -(80)/(9)) = (sqrt89)/(3) = (2)/(3) sqrt21`
Promotional Banner

Topper's Solved these Questions

  • PROPERTIES AND SOLUTIONS OF TRIANGLE

    CENGAGE PUBLICATION|Exercise Matrix match type|6 Videos

  • PROPERTIES AND SOLUTIONS OF TRIANGLE

    CENGAGE PUBLICATION|Exercise Numerical value type|22 Videos

  • PROPERTIES AND SOLUTIONS OF TRIANGLE

    CENGAGE PUBLICATION|Exercise Multiple correct answer type|24 Videos

  • PROGRESSION AND SERIES

    CENGAGE PUBLICATION|Exercise ARCHIVES (NUMERICAL VALUE TYPE )|8 Videos

  • RELATIONS AND FUNCTIONS

    CENGAGE PUBLICATION|Exercise All Questions|1119 Videos

Similar Questions

Explore conceptually related problems

In DeltaABC, P,Q, R are the feet of angle bisectors from the vertices to their opposite sides as shown in the figure. DeltaPQR is constructed If angleBAC = 120^(@) , then measusred of angleRPQ will be

The inradius in a right angled triangle with integer sides is r If r = 5, the greatest area (in sq. units) is

Let p,q,r be the sides opposite to the angles P,Q, R respectively in a triangle PQR if r^(2)sin P sin Q= pq , then the triangle is

P (2, -5), Q(1, -2) and R(4, 7) are the vertices of a triangle. Find the coordinates of the point of intersection of its side PR and the bisector of the angle Q.

In Delta ABC , right angle is at B,AB = 5cm and angle ACB =30^(@) Determine the lengths of the sides BC and AC.

A perpendicular AD is drawn on BC from the vertex A of the acute Delta ABC . Prove that AC^(2) = AB^(2) +BC^(2) -2BC.BD OR Prove that the square drawn on the opposite side of theacute angle of an acute triangle is equal to the areas of the sum of the squares drawn on its other two sides being subtracted by twice the area of the rectangle formed by one of its sides and the projection of the other side to this side.

Draw a triangle ABC and mark the midpoints E and F of two sides of triangle. bar(AB) and bar(AC) respectively. Join the point E and F as shown in the figure. Measure EF and the third side BC of the triangle. Also measure angleAEF and angleABC .

In a cyclic quadrilateral PQRS, PQ= 2 units, QR= 5 units, RS=3 units and /_P Q R=60^0, then what is the measure of SP?

ABC is a trangle whose sides are a units, b units and c units respectively and their corresponding opposite angles are A,B and C. If in the triangle ABC, the side c and angle C remain unchanged while the other sides and angles are changed slightly then the value of (da)/(cosA) + (db)/(cosB) is -

The inradius in a right angled triangle with integer sides is r If r = 4, the greatest perimeter (in units) is

CENGAGE PUBLICATION-PROPERTIES AND SOLUTIONS OF TRIANGLE-Linked comprehension type
  1. An isosceles triangle has two equal sides of length 'a' and angle betw...

    Text Solution

    |

  2. Incircle of DeltaABC touches the sides BC, AC and AB at D, E and F, re...

    Text Solution

    |

  3. Incircle of DeltaABC touches the sides BC, AC and AB at D, E and F, re...

    Text Solution

    |

  4. Incircle of DeltaABC touches the sides BC, AC and AB at D, E and F, re...

    Text Solution

    |

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

    Text Solution

    |

  6. Internal bisectors of DeltaABC meet the circumcircle at point D, E, an...

    Text Solution

    |

  7. Internal bisectors of DeltaABC meet the circumcircle at point D, E, an...

    Text Solution

    |

  8. The area of any cyclic quadrilateral ABCD is given by A^(2) = (s -a) (...

    Text Solution

    |

  9. The area of any cyclic quadrilateral ABCD is given by A^(2) = (s -a) (...

    Text Solution

    |

  10. The area of any cyclic quadrilateral ABCD is given by A^(2) = (s -a) (...

    Text Solution

    |

  11. In DeltaABC, R, r, r(1), r(2), r(3) denote the circumradius, inradius,...

    Text Solution

    |

  12. In DeltaABC, R, r, r(1), r(2), r(3) denote the circumradius, inradius,...

    Text Solution

    |

  13. In DeltaABC, R, r, r(1), r(2), r(3) denote the circumradius, inradius,...

    Text Solution

    |

  14. In DeltaABC, P,Q, R are the feet of angle bisectors from the vertices ...

    Text Solution

    |

  15. In triangleABC, P,Q, R are the feet of angle bisectors from the vertic...

    Text Solution

    |

  16. Let G be the centroid of triangle ABC and the circumcircle of triangle...

    Text Solution

    |

  17. Let G be the centroid of triangle ABC and the circumcircle of triangle...

    Text Solution

    |

  18. Let G be the centroid of triangle ABC and the circumcircle of triangle...

    Text Solution

    |

  19. The inradius in a right angled triangle with integer sides is r If r...

    Text Solution

    |

  20. The inradius in a right angled triangle with integer sides is r If r...

    Text Solution

    |