In the figure AB, CD, are extented and intersect at p.
if AB=5, BP=3, PD=2 then find CD?
a)draw the figure.
b)write the relation between PA, PB, PC, PD.
c)find CD
Answers
Answer:
see below:
Step-by-step explanation:
We join OA and drop perpendiculars OM & ON from O to CD & AB respectively.
So CM=
2
1
CD & AN=
2
1
BD.........(i) (since the perpendicular
from the centre of a circle to its chord bisects the lattar).
Also AB=AP+PB=(4+6) cm=10cm.
Now the chords AB & CD intersect at P within the given circle.
∴ PC× PD=AP× BP
⟹ PD=
PC
AP×BP
=
2
4×6
cm=12 cm.
∴ CD=PD+PC=(12+2) cm=14 cm.
∴ CM =
2
1
×14cm=7 cm and
AN=
2
1
×10 cm =5 cm (from i).
So PM=CM−PC=(7−2) cm=5 cm.
Now the quadrilateral OMPN has three of its angles =90
o
each.
∴ The quadrilateral OMPN is a rectangle.
i.e ON=PM=5 cm.
Let us consider the triangle OAN.
∠ ANO=90
o
, ON=5 cm and AN=5 cm.
∴Δ OAN is a right one with OA, which is the radius$$=r
of the circle, as its hypotenuse.
So, by Pythagoras theorem, we get OA
2
=(ON
2
+AN
2
)=(5
2
+5
2
)cm=50 cm=r
2
.
So ar.(circle)=πr
2
=π×50cm
2
=50πcm
2
.
Ans- Option C.
solution