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In the figure AB is a chord of a circle with centre O and AB is produced to C such that BC=OB .Also CO is joined and produced to meet the circle in D. If angleACD =Y ' and angle AOD=x .prove that x=3y.?
Most Upvoted Answer
?In the figure AB is a chord of a circle with centre O and AB is produ...
To prove - x = 3y
OB = BC (GIVEN)
ANGLE BOC = ANGEL BCO = y (ANGEL OPPOSITE TO EQUAL SIDE ARE EQUAL)
EXTERIOR ANGEL OBA = ANGEL BOC + ANGEL BCO = 2y
OA = OB
ANGEL OAB = ANGEL OBA =2y
EXTERIOR ANGEL AOD = ANGEL OAC + ANGEL ACO
ANGEL OAB + ANGEL BCO = 3y
Hence,
X = 3y
Proved
OB = BC (GIVEN)
ANGLE BOC = ANGEL BCO = y (ANGEL OPPOSITE TO EQUAL SIDE ARE EQUAL)
EXTERIOR ANGEL OBA = ANGEL BOC + ANGEL BCO = 2y
OA = OB
ANGEL OAB = ANGEL OBA =2y
EXTERIOR ANGEL AOD = ANGEL OAC + ANGEL ACO
ANGEL OAB + ANGEL BCO = 3y
Hence,
X = 3y
Proved
Community Answer
?In the figure AB is a chord of a circle with centre O and AB is produ...
Given:
- AB is a chord of a circle with centre O.
- AB is produced to C such that BC = OB.
- CO is joined and produced to meet the circle in D.
- Angle ACD = Y'.
- Angle AOD = X.
To prove:
X = 3Y'.
Proof:
Statement 1: Angle AOB = Angle ACB (Alternate segment theorem)
Reason: Angle at the circumference subtended by the same arc is equal.
Statement 2: Angle AOB = 2Y' (Angle in a semi-circle)
Reason: Angle in a semi-circle is always a right angle (180 degrees).
Statement 3: Angle AOC = Angle AOD (Alternate segment theorem)
Reason: Angle at the circumference subtended by the same arc is equal.
Statement 4: Angle AOC = 180 - X (Angle sum of a quadrilateral)
Reason: Sum of angles in a quadrilateral is 360 degrees.
Statement 5: Angle AOD = 180 - 2Y' (Angle sum of a triangle)
Reason: Sum of angles in a triangle is 180 degrees.
Statement 6: Angle AOC = 180 - X = 180 - 2Y'
Reason: From statements 4 and 5.
Statement 7: 180 - X = 180 - 2Y'
Reason: Transitive property.
Statement 8: X = 2Y'
Reason: Simplifying statement 7.
Statement 9: X = 3Y' (Multiplying both sides by 3)
Reason: Multiplicative property of equality.
Therefore, X = 3Y', which proves the given statement.
- AB is a chord of a circle with centre O.
- AB is produced to C such that BC = OB.
- CO is joined and produced to meet the circle in D.
- Angle ACD = Y'.
- Angle AOD = X.
To prove:
X = 3Y'.
Proof:
Statement 1: Angle AOB = Angle ACB (Alternate segment theorem)
Reason: Angle at the circumference subtended by the same arc is equal.
Statement 2: Angle AOB = 2Y' (Angle in a semi-circle)
Reason: Angle in a semi-circle is always a right angle (180 degrees).
Statement 3: Angle AOC = Angle AOD (Alternate segment theorem)
Reason: Angle at the circumference subtended by the same arc is equal.
Statement 4: Angle AOC = 180 - X (Angle sum of a quadrilateral)
Reason: Sum of angles in a quadrilateral is 360 degrees.
Statement 5: Angle AOD = 180 - 2Y' (Angle sum of a triangle)
Reason: Sum of angles in a triangle is 180 degrees.
Statement 6: Angle AOC = 180 - X = 180 - 2Y'
Reason: From statements 4 and 5.
Statement 7: 180 - X = 180 - 2Y'
Reason: Transitive property.
Statement 8: X = 2Y'
Reason: Simplifying statement 7.
Statement 9: X = 3Y' (Multiplying both sides by 3)
Reason: Multiplicative property of equality.
Therefore, X = 3Y', which proves the given statement.
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?In the figure AB is a chord of a circle with centre O and AB is produced to C such that BC=OB .Also CO is joined and produced to meet the circle in D. If angleACD =Y ' and angle AOD=x .prove that x=3y.?
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?In the figure AB is a chord of a circle with centre O and AB is produced to C such that BC=OB .Also CO is joined and produced to meet the circle in D. If angleACD =Y ' and angle AOD=x .prove that x=3y.? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about ?In the figure AB is a chord of a circle with centre O and AB is produced to C such that BC=OB .Also CO is joined and produced to meet the circle in D. If angleACD =Y ' and angle AOD=x .prove that x=3y.? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for ?In the figure AB is a chord of a circle with centre O and AB is produced to C such that BC=OB .Also CO is joined and produced to meet the circle in D. If angleACD =Y ' and angle AOD=x .prove that x=3y.?.
?In the figure AB is a chord of a circle with centre O and AB is produced to C such that BC=OB .Also CO is joined and produced to meet the circle in D. If angleACD =Y ' and angle AOD=x .prove that x=3y.? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about ?In the figure AB is a chord of a circle with centre O and AB is produced to C such that BC=OB .Also CO is joined and produced to meet the circle in D. If angleACD =Y ' and angle AOD=x .prove that x=3y.? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for ?In the figure AB is a chord of a circle with centre O and AB is produced to C such that BC=OB .Also CO is joined and produced to meet the circle in D. If angleACD =Y ' and angle AOD=x .prove that x=3y.?.
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