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Let H, I and O be the orthocentre, incentre and circumcentre of a triangle ABC If angle HAI= alpha nd angle AO= beta , Perpendicular dropped irom 0 on AC is D then which of the following is / are true
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Let H, I and O be the orthocentre, incentre and circumcentre of a tria...
H, I and O in a Triangle
In a triangle ABC, let H be the orthocentre, I be the incentre, and O be the circumcentre. We are given that angle HAI is denoted by alpha and angle AO is denoted by beta. We are also given that a perpendicular is dropped from O on AC, and the point of intersection is D. We need to determine which of the following statements are true.
Statement 1: Angle HAI = Angle DAO
To prove this statement, we need to show that angle HAI and angle DAO are equal.
Proof:
1. Draw a line segment OD perpendicular to AC from point O.
2. Since O is the circumcentre, OD is the perpendicular bisector of AC.
3. Let M be the midpoint of AC. Therefore, OM is perpendicular to AC and bisects it.
4. Since OM is the perpendicular bisector of AC, OM passes through the midpoint of AC, i.e., point D.
5. In triangle ADO, angle ADO = 90 degrees (since OD is perpendicular to AC).
6. In triangle ADO, angle DAO = 90 - alpha (angle ADO is a right angle and angle ADO = alpha).
7. In triangle ADO, angle DAO = 90 - alpha = angle HAI (given).
Therefore, statement 1 is true.
Statement 2: Angle DAO = Angle HAI + Angle OAI
To prove this statement, we need to show that angle DAO is equal to the sum of angles HAI and OAI.
Proof:
1. In triangle ADO, angle DAO = 90 - alpha.
2. In triangle ADO, angle HAI = alpha (given).
3. In triangle ADO, angle OAI = 90 degrees (angle OAI = 90 - alpha, as angle HAI + angle OAI = 90 degrees).
4. Therefore, angle DAO = 90 - alpha = angle HAI + angle OAI.
Therefore, statement 2 is true.
Conclusion:
From the above proofs, we can conclude that both statement 1 and statement 2 are true.
In a triangle ABC, let H be the orthocentre, I be the incentre, and O be the circumcentre. We are given that angle HAI is denoted by alpha and angle AO is denoted by beta. We are also given that a perpendicular is dropped from O on AC, and the point of intersection is D. We need to determine which of the following statements are true.
Statement 1: Angle HAI = Angle DAO
To prove this statement, we need to show that angle HAI and angle DAO are equal.
Proof:
1. Draw a line segment OD perpendicular to AC from point O.
2. Since O is the circumcentre, OD is the perpendicular bisector of AC.
3. Let M be the midpoint of AC. Therefore, OM is perpendicular to AC and bisects it.
4. Since OM is the perpendicular bisector of AC, OM passes through the midpoint of AC, i.e., point D.
5. In triangle ADO, angle ADO = 90 degrees (since OD is perpendicular to AC).
6. In triangle ADO, angle DAO = 90 - alpha (angle ADO is a right angle and angle ADO = alpha).
7. In triangle ADO, angle DAO = 90 - alpha = angle HAI (given).
Therefore, statement 1 is true.
Statement 2: Angle DAO = Angle HAI + Angle OAI
To prove this statement, we need to show that angle DAO is equal to the sum of angles HAI and OAI.
Proof:
1. In triangle ADO, angle DAO = 90 - alpha.
2. In triangle ADO, angle HAI = alpha (given).
3. In triangle ADO, angle OAI = 90 degrees (angle OAI = 90 - alpha, as angle HAI + angle OAI = 90 degrees).
4. Therefore, angle DAO = 90 - alpha = angle HAI + angle OAI.
Therefore, statement 2 is true.
Conclusion:
From the above proofs, we can conclude that both statement 1 and statement 2 are true.
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Let H, I and O be the orthocentre, incentre and circumcentre of a tria...
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Let H, I and O be the orthocentre, incentre and circumcentre of a triangle ABC If angle HAI= alpha nd angle AO= beta , Perpendicular dropped irom 0 on AC is D then which of the following is / are true Related: Perpendicular bisector - Triangle and Its Properties, Mathematics, CBSE Class 7?
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Let H, I and O be the orthocentre, incentre and circumcentre of a triangle ABC If angle HAI= alpha nd angle AO= beta , Perpendicular dropped irom 0 on AC is D then which of the following is / are true Related: Perpendicular bisector - Triangle and Its Properties, Mathematics, CBSE Class 7? for Class 7 2024 is part of Class 7 preparation. The Question and answers have been prepared according to the Class 7 exam syllabus. Information about Let H, I and O be the orthocentre, incentre and circumcentre of a triangle ABC If angle HAI= alpha nd angle AO= beta , Perpendicular dropped irom 0 on AC is D then which of the following is / are true Related: Perpendicular bisector - Triangle and Its Properties, Mathematics, CBSE Class 7? covers all topics & solutions for Class 7 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let H, I and O be the orthocentre, incentre and circumcentre of a triangle ABC If angle HAI= alpha nd angle AO= beta , Perpendicular dropped irom 0 on AC is D then which of the following is / are true Related: Perpendicular bisector - Triangle and Its Properties, Mathematics, CBSE Class 7?.
Let H, I and O be the orthocentre, incentre and circumcentre of a triangle ABC If angle HAI= alpha nd angle AO= beta , Perpendicular dropped irom 0 on AC is D then which of the following is / are true Related: Perpendicular bisector - Triangle and Its Properties, Mathematics, CBSE Class 7? for Class 7 2024 is part of Class 7 preparation. The Question and answers have been prepared according to the Class 7 exam syllabus. Information about Let H, I and O be the orthocentre, incentre and circumcentre of a triangle ABC If angle HAI= alpha nd angle AO= beta , Perpendicular dropped irom 0 on AC is D then which of the following is / are true Related: Perpendicular bisector - Triangle and Its Properties, Mathematics, CBSE Class 7? covers all topics & solutions for Class 7 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let H, I and O be the orthocentre, incentre and circumcentre of a triangle ABC If angle HAI= alpha nd angle AO= beta , Perpendicular dropped irom 0 on AC is D then which of the following is / are true Related: Perpendicular bisector - Triangle and Its Properties, Mathematics, CBSE Class 7?.
Solutions for Let H, I and O be the orthocentre, incentre and circumcentre of a triangle ABC If angle HAI= alpha nd angle AO= beta , Perpendicular dropped irom 0 on AC is D then which of the following is / are true Related: Perpendicular bisector - Triangle and Its Properties, Mathematics, CBSE Class 7? in English & in Hindi are available as part of our courses for Class 7.
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