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In triangle ABC ,the internal bisectors of angle b and angle c meet at O. Prove that OA is also the internal bisector of angle A?
Verified Answer
In triangle ABC ,the internal bisectors of angle b and angle c meet at...
Given : In a triangle ABC the internal bisectors of Angle B and angle C meet at O
To find : prove that OA is the internal bisector of angle A
Solution:
Lets draw a line from A passing through O meeting BC at D
in Δ ABD
BO is angle bisector
Hence AB / BD = AO/OD ( internal bisector theorem )
in Δ ACD
CO is angle bisector
Hence AC / CD = AO/OD ( internal bisector theorem )
Equating Both
AB / BD = AC / CD
=> AB/ AC = BD/CD
Hence AD is angle bisector of ∠A in Δ ABC ( Converse of internal bisector theorem )
O lies on AD
Hence OA is internal bisector of ∠A
QED
Hence proved
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In triangle ABC ,the internal bisectors of angle b and angle c meet at...
In triangle ABC ,the internal bisectors of angle b and angle c meet at O. Prove that OA is also the internal bisector of angle A?
Community Answer
In triangle ABC ,the internal bisectors of angle b and angle c meet at...
Given:
Triangle ABC
Internal bisectors of angle B and angle C meet at point O
To prove:
OA is the internal bisector of angle A
Proof:
Step 1: Construct the diagram
Draw a triangle ABC on a paper or use a geometry software to construct the triangle.
Step 2: Identify the given information
In triangle ABC, we are given that the internal bisectors of angle B and angle C meet at point O.
Step 3: Identify the angles to be proved
We need to prove that OA is the internal bisector of angle A.
Step 4: Definition of an internal bisector
An internal bisector of an angle is a line or ray that divides the angle into two equal angles.
Step 5: Definition of angle bisector theorem
According to the angle bisector theorem, the internal bisector of an angle divides the opposite side of the triangle into two segments that are proportional to the other two sides of the triangle.
Step 6: Apply the angle bisector theorem
Since the internal bisector of angle B intersects side AC at point O, we can apply the angle bisector theorem to triangle ABC.
The angle bisector theorem states that the ratio of the lengths of the segments formed by the angle bisector on the opposite side is equal to the ratio of the lengths of the other two sides.
In triangle ABC, let the lengths of sides AB, BC, and AC be denoted by a, b, and c, respectively. Let the length of segment AO be x, and the length of segment OC be y.
According to the angle bisector theorem,
x/y = AB/BC
Step 7: Apply the angle bisector theorem again
Similarly, since the internal bisector of angle C intersects side AB at point O, we can apply the angle bisector theorem to triangle ABC again.
In triangle ABC, let the lengths of sides AB, BC, and AC be denoted by a, b, and c, respectively. Let the length of segment AO be x, and the length of segment OB be z.
According to the angle bisector theorem,
x/z = AC/BC
Step 8: Equate the two ratios
Since both ratios are equal to AB/BC, we can equate them:
x/y = x/z
Step 9: Simplify the equation
Cross-multiplying the equation, we get:
xz = xy
Step 10: Conclusion
Since both sides of the equation are equal, we can conclude that OA is the internal bisector of angle A.
Hence, the given statement is proved.
Triangle ABC
Internal bisectors of angle B and angle C meet at point O
To prove:
OA is the internal bisector of angle A
Proof:
Step 1: Construct the diagram
Draw a triangle ABC on a paper or use a geometry software to construct the triangle.
Step 2: Identify the given information
In triangle ABC, we are given that the internal bisectors of angle B and angle C meet at point O.
Step 3: Identify the angles to be proved
We need to prove that OA is the internal bisector of angle A.
Step 4: Definition of an internal bisector
An internal bisector of an angle is a line or ray that divides the angle into two equal angles.
Step 5: Definition of angle bisector theorem
According to the angle bisector theorem, the internal bisector of an angle divides the opposite side of the triangle into two segments that are proportional to the other two sides of the triangle.
Step 6: Apply the angle bisector theorem
Since the internal bisector of angle B intersects side AC at point O, we can apply the angle bisector theorem to triangle ABC.
The angle bisector theorem states that the ratio of the lengths of the segments formed by the angle bisector on the opposite side is equal to the ratio of the lengths of the other two sides.
In triangle ABC, let the lengths of sides AB, BC, and AC be denoted by a, b, and c, respectively. Let the length of segment AO be x, and the length of segment OC be y.
According to the angle bisector theorem,
x/y = AB/BC
Step 7: Apply the angle bisector theorem again
Similarly, since the internal bisector of angle C intersects side AB at point O, we can apply the angle bisector theorem to triangle ABC again.
In triangle ABC, let the lengths of sides AB, BC, and AC be denoted by a, b, and c, respectively. Let the length of segment AO be x, and the length of segment OB be z.
According to the angle bisector theorem,
x/z = AC/BC
Step 8: Equate the two ratios
Since both ratios are equal to AB/BC, we can equate them:
x/y = x/z
Step 9: Simplify the equation
Cross-multiplying the equation, we get:
xz = xy
Step 10: Conclusion
Since both sides of the equation are equal, we can conclude that OA is the internal bisector of angle A.
Hence, the given statement is proved.
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In triangle ABC ,the internal bisectors of angle b and angle c meet at O. Prove that OA is also the internal bisector of angle A?
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In triangle ABC ,the internal bisectors of angle b and angle c meet at O. Prove that OA is also the internal bisector of angle A? 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 triangle ABC ,the internal bisectors of angle b and angle c meet at O. Prove that OA is also the internal bisector of angle A? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In triangle ABC ,the internal bisectors of angle b and angle c meet at O. Prove that OA is also the internal bisector of angle A?.
In triangle ABC ,the internal bisectors of angle b and angle c meet at O. Prove that OA is also the internal bisector of angle A? 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 triangle ABC ,the internal bisectors of angle b and angle c meet at O. Prove that OA is also the internal bisector of angle A? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In triangle ABC ,the internal bisectors of angle b and angle c meet at O. Prove that OA is also the internal bisector of angle A?.
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