Home
Class 9
MATHS
The diagonals AC and BD of a cyclic quad...

The diagonals `AC` and `BD` of a cyclic quadrilateral `ABCD` interest at right angles at E (figure). A line l drawn through E and perpendicular to AB meets CD at F. Prove that F is the mid-point of CD.

Text Solution

Verified by Experts

ABCD is a cyclic quadrilateral in which diagonals AC and BD intersect in E at right angles , l is a line through E and perpendicular to AB meets CD in F.
We have `angle AEB=90^@ " "(because "the diagonals are at right angles")`
`rArrangle1_angle2 =90^@`
`angle1 =90^@-angle2` .....(1)
`rArrangle1 =90^@-angle2" "(because EMbotAB)`
Again `angle EMB=90^@`
In `Delta EMB`, we have
`angle2+angle4=90^@`
`rArrangle4=90^@-anlge2` .......(2)
From eqs.(1) and (2) , we get
`angle1=angle4`
But `angle1 =angle3` (vertically opposite angles)
`therefore angle3=angle4`
Also, ` angle4=angle5` (angles in the same segment of a circle are equal)
`rArr angle3=angle5`
`rArrCF=EF` (sides opposite to equal angles of a triangle are equal)
Similarly , `DF=EF`
`rArrCF=DF`
Hence, F is the mid-point of CD.
Promotional Banner

Topper's Solved these Questions

  • CIRCLE

    NAGEEN PRAKASHAN|Exercise Exercise 10a|22 Videos

  • CIRCLE

    NAGEEN PRAKASHAN|Exercise Exercise 10b|19 Videos

  • AREA OF PARALLELOGRAMS AND TRIANGLES

    NAGEEN PRAKASHAN|Exercise Revision Exercise (long Answer Question)|5 Videos

  • CO-ORDINATE GEOMETRY

    NAGEEN PRAKASHAN|Exercise Exercise|8 Videos

Similar Questions

Explore conceptually related problems

AB and CD are the parallel sides of a trapeziuml. E is the mid-point of AD. A line through E and parallel to side AB meets the line BC at point F. Prove that F is the mid-point of BC.

In cyclic quadrilateral ABCD, AD|| BC. Prove that AB = CD.

In Fig 9.25 ,diagonals AC and BD of quadrilateral ABCD intersect at O such that OB= OD.If AB=CD, then show that

In the figure, diagonal AC of a quadrilateral ABCD bisects the angles A and C. Prove that AB = AD and CB = CD.

In the figure, diagonals AC and BD of quadrilateral ABCD intersect at O such that OB = OD. If AB = CD, then show that ar( Delta DOC) = ar( Delta AOB)

In Figure,diagonal AC of a quadrilateral ABCD bisects the angles A and C. Prove that AB=AD and CB=CD

In the figur e, diagonals AC and BD of quadrilateral A BCD intersect at 0 such that OB = OD. If AB = CD, then show that ar( Delta DCB) = ar( Delta ACB)

In a right triangle ABC, right angled at A, AD is drawn perpendicular to BC. Prove that: AB^(2)-BD^(2)= AC^(2)-CD^(2)

ABCD is a quadrilateral.Is AB+BC+CD+DA>AC+BD?

ABCD is a trapezium in which A B\ ||\ D C , BD is a diagonal and E is the mid-point of AD. A line is drawn through E parallel to AB intersecting BC at F (see Fig. 8.30). Show that F is the mid-point of BC.