Distance between the Points
Consider a semicircle of radius 1 on the diameter A B where the center is O .
A point C divides A O in the ratio 2 : 1 .
A line that is perpendicular to A O passing through C cuts the semicircle at E .
Another line O L passing through O and is perpendicular to A E intersects C E at H .
Find the value of E H .
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3 solutions
We first get
A
C
=
3
2
,
C
O
=
3
1
and
E
C
=
3
2
2
using
Pythagoras Theorem
on
△
E
C
O
.
We observe that
A
L
=
L
E
since
△
A
O
E
is isosceles and
C
H
=
3
2
2
−
3
E
H
.
We apply
Menelaus Theorem
for
△
E
C
A
with
L
O
as the transversal.
So,
C
O
A
O
×
E
H
C
H
×
A
L
L
E
=
1
[
Actually it should be
−
1
but it doesn't affect the problem as the minus sign is just to indicate that the point
O
does not lie between
A
C
]
.
⇒
3
1
1
×
E
H
C
H
×
A
L
L
E
=
1
.
⇒
E
H
C
H
=
3
1
.
⇒
3
E
H
2
2
−
3
E
H
=
3
1
.
Therefore after solving we get
E
H
=
2
1
.
Hey how was you able to make a cut ?
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latex thinking.\require{cancel} mention this at the beginning of your statement where u want to make the cut.then \cancel{3}.this will be represented as 3 .
