Question

hypotenuse.
(3) In the parallelogram ABCD, the line drawn through a point P on AB,
parallel to BC, meets AC at Q. The line through Q, parallel to AB meets AD at R.

Prove that
AP AR
----- = -----
PB RD​

Answer 1

Answer:

The angles of quadrilateral are in the ratio 3 : 5 : 9 : quadrilateral.

Sol: Let the angles of the quadrilateral be 3x, 5x, 9x and 13x.

∵ Sum of all the angles of quadrilateral = 360°

∴ 3x + 5x + 9x + 13x = 360°

⇒ 30x = 360°

∴ 3x = 3 × 12° = 36°

5x = 5 × 12°= 60°

9x = 9 × 12° = 108°

13x = 13 × 12° = 156°

⇒ The required angles of the quadrilateral are 36°, 60°, 108° and 156°.

Q2. If the diagonals of a parallelogram are equal, then show that it is a rectangle.

Sol: A parallelogram ABCD such that AC = BD In ΔABC and ΔDCB,

AC = DB

[Given]

AB = DC

[Opposite sides of a parallelogram]

BC = CB

[Common]

ΔABC = ΔDCB

[SSS criteria]

∴Their corresponding parts are equal.

⇒∠ABC = ∠DCB

...(1)

∵AB || DC and BC is a transversal.

[∵ ABCD is a parallelogram]

∴∠ABC + ∠DCB = 180°

...(2)

From (1) and (2), we have

∠ABC = ∠DCB = 90°

i.e. ABCD is parallelogram having an angle equal to 90°.

∴ABCD is a rectangle.

Answer 2

Answer:

In ΔABC, RQ the parallel line of BC, divides AB and BC in the same ratio When we consider the relation A R R D = A Q Q C , A P P B = A Q A C = A R R D ARRD=AQQC,APPB=AQAC=ARRD A R R D = A Q Q C ARRD=AQQC A P P B = A Q A C = A R R D APPB=AQAC=ARRD ∴ A P P B = A R R

Step-by-step explanation:

your answer in. image