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p is a point on side BC of triangle ABC such that PL is perpendicular on AB and PM perpendicular on AC and PL is equal to PM . show that AP bisects angle BAC.
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p is a point on side BC of triangle ABC such that PL is perpendicular ...
ABCD is a quadrilateral in which AD = BC and ∠DAB = ∠CBA (see figure). Prove that:
(i) ∆ABD ≅ ∆BAC
(ii) BD = AC
(iii) ∠ABD = ∠BAC.
This question is part of UPSC exam. View all Class 7 courses
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p is a point on side BC of triangle ABC such that PL is perpendicular ...
Given:
- Triangle ABC with point P on side BC such that PL is perpendicular to AB and PM is perpendicular to AC.
- PL = PM
To prove:
- AP bisects angle BAC
Proof:
Step 1: Draw a diagram
- Draw triangle ABC with point P on side BC.
- Draw perpendiculars PL and PM from P to AB and AC respectively.
- Label the intersection of AP with BC as Q.
Step 2: Establish the given information
- Given that PL = PM, which means triangle PML is an isosceles triangle.
- Therefore, angles PLM and PML are equal.
Step 3: Establish congruence
- Triangles APL and AMP are congruent by RHS congruence criterion.
- Side AP is common.
- PL = PM (given)
- Angle APL = Angle AMP = 90° (perpendicularity)
- Therefore, AP = AM and angle APL = angle AMP.
Step 4: Prove angle APQ = angle APM
- Triangles APL and AQM are congruent by RHS congruence criterion.
- Side AP is common.
- PL = QM (perpendicularity and definition of Q)
- Angle APL = Angle AQM = 90° (perpendicularity)
- Therefore, AP = AQ and angle APL = angle AQM.
Step 5: Conclude that AP bisects angle BAC
- Since AP = AM and angle APL = angle AMP, and AP = AQ and angle APL = angle AQM, we can conclude that angle APQ = angle APM.
- Therefore, AP bisects angle BAC.
Conclusion:
- We have proved that if PL is perpendicular to AB and PM is perpendicular to AC, with PL = PM, then AP bisects angle BAC.
- Triangle ABC with point P on side BC such that PL is perpendicular to AB and PM is perpendicular to AC.
- PL = PM
To prove:
- AP bisects angle BAC
Proof:
Step 1: Draw a diagram
- Draw triangle ABC with point P on side BC.
- Draw perpendiculars PL and PM from P to AB and AC respectively.
- Label the intersection of AP with BC as Q.
Step 2: Establish the given information
- Given that PL = PM, which means triangle PML is an isosceles triangle.
- Therefore, angles PLM and PML are equal.
Step 3: Establish congruence
- Triangles APL and AMP are congruent by RHS congruence criterion.
- Side AP is common.
- PL = PM (given)
- Angle APL = Angle AMP = 90° (perpendicularity)
- Therefore, AP = AM and angle APL = angle AMP.
Step 4: Prove angle APQ = angle APM
- Triangles APL and AQM are congruent by RHS congruence criterion.
- Side AP is common.
- PL = QM (perpendicularity and definition of Q)
- Angle APL = Angle AQM = 90° (perpendicularity)
- Therefore, AP = AQ and angle APL = angle AQM.
Step 5: Conclude that AP bisects angle BAC
- Since AP = AM and angle APL = angle AMP, and AP = AQ and angle APL = angle AQM, we can conclude that angle APQ = angle APM.
- Therefore, AP bisects angle BAC.
Conclusion:
- We have proved that if PL is perpendicular to AB and PM is perpendicular to AC, with PL = PM, then AP bisects angle BAC.
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p is a point on side BC of triangle ABC such that PL is perpendicular on AB and PM perpendicular on AC and PL is equal to PM . show that AP bisects angle BAC. Related: Congruence Among Right angled Triangle:RHS Congruence Criterion, Congruence of Triangles, Class 7
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p is a point on side BC of triangle ABC such that PL is perpendicular on AB and PM perpendicular on AC and PL is equal to PM . show that AP bisects angle BAC. Related: Congruence Among Right angled Triangle:RHS Congruence Criterion, Congruence of Triangles, 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 p is a point on side BC of triangle ABC such that PL is perpendicular on AB and PM perpendicular on AC and PL is equal to PM . show that AP bisects angle BAC. Related: Congruence Among Right angled Triangle:RHS Congruence Criterion, Congruence of Triangles, Class 7 covers all topics & solutions for Class 7 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for p is a point on side BC of triangle ABC such that PL is perpendicular on AB and PM perpendicular on AC and PL is equal to PM . show that AP bisects angle BAC. Related: Congruence Among Right angled Triangle:RHS Congruence Criterion, Congruence of Triangles, Class 7.
p is a point on side BC of triangle ABC such that PL is perpendicular on AB and PM perpendicular on AC and PL is equal to PM . show that AP bisects angle BAC. Related: Congruence Among Right angled Triangle:RHS Congruence Criterion, Congruence of Triangles, 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 p is a point on side BC of triangle ABC such that PL is perpendicular on AB and PM perpendicular on AC and PL is equal to PM . show that AP bisects angle BAC. Related: Congruence Among Right angled Triangle:RHS Congruence Criterion, Congruence of Triangles, Class 7 covers all topics & solutions for Class 7 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for p is a point on side BC of triangle ABC such that PL is perpendicular on AB and PM perpendicular on AC and PL is equal to PM . show that AP bisects angle BAC. Related: Congruence Among Right angled Triangle:RHS Congruence Criterion, Congruence of Triangles, Class 7.
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