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A point P is in the interior of angle BAC, such that P lies on the bisector of angle BAC. What can be said about the distance of PM if PN = 2cm where PM and PN are perpendiculars from P on the lines BA and AC?
- a)PM = 2 PN
- b)PM = 3 PN
- c)PM =2 cm
- d)PM =3.5 cm
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
A point P is in the interior of angle BAC, such that P lies on the bis...
In Δ PAM & Δ PAN, angle (PAM) = angle (PAN) (Since AP bisects the angle (BAC)) angle (AMP) = angle (ANP) = 900
PA = PA (common)
By AAS congruence, Δ PAM and Δ PAN are congruent.
MP = NP = 2 cm
PA = PA (common)
By AAS congruence, Δ PAM and Δ PAN are congruent.
MP = NP = 2 cm
Most Upvoted Answer
A point P is in the interior of angle BAC, such that P lies on the bis...
Understanding the Geometry
When a point P lies on the angle bisector of angle BAC, it creates a unique relationship between the distances from P to the sides of the angle.
Key Distances
- PM and PN are the perpendicular distances from point P to lines BA and AC, respectively.
- Given that PN = 2 cm, we need to establish the relationship between PM and PN.
Using Angle Bisector Theorem
The Angle Bisector Theorem states that the ratio of the lengths of the segments created by the angle bisector is equal to the ratio of the other two sides of the triangle. Consequently, since P lies on the bisector:
- The distances from P to the sides are proportional.
- Therefore, if the distances are perpendiculars to the angle sides, the relationship can be described as:
PM / PN = k (where k is a constant depending on the angles).
Calculating PM
Since we know PN = 2 cm, we can express PM based on this distance. The angle bisector creates a specific proportional relationship:
- PM = k * PN
- By the properties of angle bisectors and the inner angles, we find that k = 1 (as both sides are equal in a symmetric setup).
However, this is simplified in many cases, leading to a specific case:
- In our given problem, particularly with the ratio and the distances being equal, we can conclude that PM will also equal 2 cm because it maintains the geometric properties of angle bisectors.
Conclusion
Thus, since PM = 2 cm, the correct answer is:
- c) PM = 2 cm
When a point P lies on the angle bisector of angle BAC, it creates a unique relationship between the distances from P to the sides of the angle.
Key Distances
- PM and PN are the perpendicular distances from point P to lines BA and AC, respectively.
- Given that PN = 2 cm, we need to establish the relationship between PM and PN.
Using Angle Bisector Theorem
The Angle Bisector Theorem states that the ratio of the lengths of the segments created by the angle bisector is equal to the ratio of the other two sides of the triangle. Consequently, since P lies on the bisector:
- The distances from P to the sides are proportional.
- Therefore, if the distances are perpendiculars to the angle sides, the relationship can be described as:
PM / PN = k (where k is a constant depending on the angles).
Calculating PM
Since we know PN = 2 cm, we can express PM based on this distance. The angle bisector creates a specific proportional relationship:
- PM = k * PN
- By the properties of angle bisectors and the inner angles, we find that k = 1 (as both sides are equal in a symmetric setup).
However, this is simplified in many cases, leading to a specific case:
- In our given problem, particularly with the ratio and the distances being equal, we can conclude that PM will also equal 2 cm because it maintains the geometric properties of angle bisectors.
Conclusion
Thus, since PM = 2 cm, the correct answer is:
- c) PM = 2 cm
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Community Answer
A point P is in the interior of angle BAC, such that P lies on the bis...
In Δ PAM & Δ PAN, angle (PAM) = angle (PAN) (Since AP bisects the angle (BAC)) angle (AMP) = angle (ANP) = 900
PA = PA (common)
By AAS congruence, Δ PAM and Δ PAN are congruent.
MP = NP = 2 cm
PA = PA (common)
By AAS congruence, Δ PAM and Δ PAN are congruent.
MP = NP = 2 cm
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A point P is in the interior of angle BAC, such that P lies on the bisector of angle BAC. What can be said about the distance of PM if PN =2cm where PM and PN are perpendiculars from P on the lines BA and AC?a)PM = 2 PNb)PM = 3 PNc)PM =2 cmd)PM =3.5 cmCorrect answer is option 'C'. Can you explain this answer? for JEE 2026 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A point P is in the interior of angle BAC, such that P lies on the bisector of angle BAC. What can be said about the distance of PM if PN =2cm where PM and PN are perpendiculars from P on the lines BA and AC?a)PM = 2 PNb)PM = 3 PNc)PM =2 cmd)PM =3.5 cmCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2026 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A point P is in the interior of angle BAC, such that P lies on the bisector of angle BAC. What can be said about the distance of PM if PN =2cm where PM and PN are perpendiculars from P on the lines BA and AC?a)PM = 2 PNb)PM = 3 PNc)PM =2 cmd)PM =3.5 cmCorrect answer is option 'C'. Can you explain this answer?.
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