Explanation
Given, ABCD is a cyclic quadrilateral. AC is its diagonal. ∠BAD=25o,∠BCD=80o.
Here, ABCD is a cyclic quadrilateral.
∴∠BCD+∠BAD=180o (since the sum of the opposite angles of a cyclic quadrilateral is 180o)
⇒∠BAD=180o−∠BCD=180o−80o=100o.
But ∠BAD=∠BAC+∠CAD
∴∠BAC+∠CAD=100o⇒∠CAD=100o−∠BAC=100o−25o=75o.
Hence, option C is correct.
Since AB is diameter,
∠AEB=90∘ ...[Angle formed in a semi circle].
In △AEB
∠ABE=180−∠EAB−∠AEB ...[Angle sum property]
∠ABE=180–65−90=25∘.
Given, ED∥AB.
⟹∠DEB=∠EBA=25∘ ...[Alternate interior angles].
Since EDCB is a cyclic quadrilateral opposite angles are supplementary
Then, ∠DEB+∠DCB=180∘
⟹∠DCB=180–25=155∘.
Hence, option D is correct.
Since AB is diameter, ∠AEB=90∘.
In △AEB,
∠ABE+∠EAB+∠AEB=180o ...[Angle sum property]
⟹ ∠ABE=180o−∠EAB−∠AEB
⟹ ∠ABE=180o–65o−90o=25∘.
Given, ED∥AB,
⟹ ∠DEB=∠EBA=25∘ ....[Alternate interior angles].
Since EDCB is a cyclic quadrilateral,
∠EAB+∠EDB=180∘ ...[Opposite angles of cyclic quadrilateral are supplementary]
⟹ ∠EDB=180o–65o=115∘.
In △EDB,
∠EBD+∠DEB+∠BDE=180o
⟹ ∠EBD=180−∠DEB−∠BDE
⟹ ∠ABE=180–25−115=40∘.
Step - 1: Verify, If we join any points on a circle we get a diameter of the circle
A Chord is a line segment that joins any two points of the circle.
The endpoints of this line segments lie on the circumference of the circle.
∴Option (A) is false
Step - 2: Verify, A diameter of a circle contains the center of the circle
Any interval joining two points on the circle and passing through the center is called a
diameter of the circle.
∴Option (B) is True
Step - 3: Verify, A semicircle is an arc
The arc of a circle consists of two points on the circle and all of the points on the circle that lie
between those two points.
It's like a segment that was wrapped partway around a circle.
An arc whose measure equals 180 degrees is called a semicircle since it divides the circle in two
∴Option (C) is True
Step - 4: Verify, the length of a circle is called its circumference
A Circle is a round closed figure where all its boundary points are equidistant from a fixed point
called the center.
The two important metrics of a circle is the area of a circle and the circumference of a circle.
∴Option (D) is True
Hence, option A is correct as it is false
Let AB,PQ be two chords, OC and OR be their distance from center.
Given
OC=OR
We know that BC=AC and PR=SR because perpendicular line from center bisects the chord.
In △OBC
BC2=r2–OC2–(1)
In △OPR
OP2=RP2+OR2
PR2=r2–OC2–(2)
(1)=(2)
⟹BC=PR⟹2BC=2PR
AB=PQ
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