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AB and AC are two equal chords of a circle whose center is O. If AB ⊥ OD and OE ⊥ AC, then,
- a)∆ABE is an isosceles triangle
- b)∆ADE is an equilateral triangle
- c)∆ADC is an isosceles triangle
- d)∆ADE is an isosceles triangle
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
AB and AC are two equal chords of a circle whose center is O. If AB&pe...
In ∆ODE,
OD = OE
∴ ∠ODE = ∠OED ……(i)
Now,
∠ODA = ∠ODE = 90° ….(ii)
Subtracting eq.(i) from (ii), we get
∠ODA – ∠ODE = ∠OEA – ∠OED
∠ADE = ∠AED
∴ AD = AE ⇒ ∆ADE is an isosceles triangle.
OD = OE
∴ ∠ODE = ∠OED ……(i)
Now,
∠ODA = ∠ODE = 90° ….(ii)
Subtracting eq.(i) from (ii), we get
∠ODA – ∠ODE = ∠OEA – ∠OED
∠ADE = ∠AED
∴ AD = AE ⇒ ∆ADE is an isosceles triangle.
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AB and AC are two equal chords of a circle whose center is O. If AB&pe...
Understanding the Circle and Chords
In this problem, we have a circle with center O, and two equal chords AB and AC. Since AB and AC are equal chords, their perpendicular distances from the center O will also be equal.
Properties of Perpendiculars from Center
- Given that AB is perpendicular to OD, and OE is perpendicular to AC, we can conclude:
- OD and OE are radii of the circle.
- The perpendicular from the center to the chord bisects the chord.
This implies that D is the midpoint of AB and E is the midpoint of AC.
Triangles ABE and ADE
- Since AB = AC, and both are bisected at D and E:
- AD = AE (as both are radii OP and OQ respectively).
Based on this information, we can analyze triangle ADE.
Isosceles Triangle ADE
- Triangle ADE has two sides (AD and AE) that are equal:
- Therefore, triangle ADE is an isosceles triangle.
Conclusion
Considering the above properties and analysis, the correct answer is option D: ADE is an isosceles triangle. This conclusion arises from the equality of the two chords and the nature of the triangles formed by the midpoints and the circle's center.
In this problem, we have a circle with center O, and two equal chords AB and AC. Since AB and AC are equal chords, their perpendicular distances from the center O will also be equal.
Properties of Perpendiculars from Center
- Given that AB is perpendicular to OD, and OE is perpendicular to AC, we can conclude:
- OD and OE are radii of the circle.
- The perpendicular from the center to the chord bisects the chord.
This implies that D is the midpoint of AB and E is the midpoint of AC.
Triangles ABE and ADE
- Since AB = AC, and both are bisected at D and E:
- AD = AE (as both are radii OP and OQ respectively).
Based on this information, we can analyze triangle ADE.
Isosceles Triangle ADE
- Triangle ADE has two sides (AD and AE) that are equal:
- Therefore, triangle ADE is an isosceles triangle.
Conclusion
Considering the above properties and analysis, the correct answer is option D: ADE is an isosceles triangle. This conclusion arises from the equality of the two chords and the nature of the triangles formed by the midpoints and the circle's center.
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AB and AC are two equal chords of a circle whose center is O. If AB⊥OD and OE ⊥AC,then,a)ABE is an isosceles triangleb)ADE is an equilateral trianglec)ADC is an isosceles triangled)ADE is an isosceles triangleCorrect answer is option 'D'. Can you explain this answer? for Class 9 2025 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about AB and AC are two equal chords of a circle whose center is O. If AB⊥OD and OE ⊥AC,then,a)ABE is an isosceles triangleb)ADE is an equilateral trianglec)ADC is an isosceles triangled)ADE is an isosceles triangleCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for Class 9 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for AB and AC are two equal chords of a circle whose center is O. If AB⊥OD and OE ⊥AC,then,a)ABE is an isosceles triangleb)ADE is an equilateral trianglec)ADC is an isosceles triangled)ADE is an isosceles triangleCorrect answer is option 'D'. Can you explain this answer?.
AB and AC are two equal chords of a circle whose center is O. If AB⊥OD and OE ⊥AC,then,a)ABE is an isosceles triangleb)ADE is an equilateral trianglec)ADC is an isosceles triangled)ADE is an isosceles triangleCorrect answer is option 'D'. Can you explain this answer? for Class 9 2025 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about AB and AC are two equal chords of a circle whose center is O. If AB⊥OD and OE ⊥AC,then,a)ABE is an isosceles triangleb)ADE is an equilateral trianglec)ADC is an isosceles triangled)ADE is an isosceles triangleCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for Class 9 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for AB and AC are two equal chords of a circle whose center is O. If AB⊥OD and OE ⊥AC,then,a)ABE is an isosceles triangleb)ADE is an equilateral trianglec)ADC is an isosceles triangled)ADE is an isosceles triangleCorrect answer is option 'D'. Can you explain this answer?.
Solutions for AB and AC are two equal chords of a circle whose center is O. If AB⊥OD and OE ⊥AC,then,a)ABE is an isosceles triangleb)ADE is an equilateral trianglec)ADC is an isosceles triangled)ADE is an isosceles triangleCorrect answer is option 'D'. Can you explain this answer? in English & in Hindi are available as part of our courses for Class 9.
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