Ex 9.1, 3

Transcript

Ex 9.1, 3 Frame the converse for each of the propositions in Questions 1–12. Then, determine if each of the two statements is true or not. Justify the true statements and give a counterexample for each false statement. Given any ΔABC, let us bisect the angles at B and C. The bisectors meet at the incentre I of the triangle. Now extend the bisectors beyond I till they meet the opposite sides at E and F respectively, as shown. Proposition: If AB = AC, then IE = IF. Let’s write Proposition and Converse first Proposition: If AB = AC, then IE = IF Converse: If IE = IF, then AB = AC Checking if Proposition is true Proposition: If AB = AC, then IE = IF Let’s prove this Given: ∆ ABC where BE is angle bisector of ∠ B, i.e. ∠ ABE = ∠ CBE = 𝟏/𝟐∠ ABC CF is angle bisector of ∠ C, i.e. ∠ ACF = ∠ CBE = 𝟏/𝟐∠ ACB And, AB = AC To prove: IE = IF Proof: Since AB = AC And we know that angles opposite equal sides are equal ∴ ∠ ABC = ∠ ACB Thus, we can write 1/2 × ∠ ABC = 1/2 × ∠ ACB ∠ CBE = ∠ BCF In ∆ BCE & ∆CBF ∠ CBE = ∠ BCF ∠ BCE = ∠ CBF BC = BC ∴ ∆ BCE ≅ ∆CBF Thus, by CPCT BE = CF Now, in ∆ BIC ∠ IBC = ∠ ICB And, sides opposite equal angles are equal ∴ BI = CI From (2) BE = CF BI + IE = CI + IF From (3): BI = CI BI + IE = BI + IF IE = IF Hence proved Thus, Proposition is true Checking if Converse is true Converse: If IE = IF, then AB = AC This is also true sd Thus, by CPCT BE = CF Now, in ∆ BIC ∠ IBC = ∠ ICB And, sides opposite equal angles are equal ∴ BI = CI From (2) BE = CF BI + IE = CI + IF From (3): BI = CI BI + IE = BI + IF IE = IF Hence proved Thus, Proposition is true Checking if Converse is true Converse: If IE = IF, then AB = AC We take a counterexample here Let’s consider a triangle with angles ∠ A = 60°, ∠ B = 90°, ∠ C = 30° Since ∠ B ≠ ∠ C, ∴ AB ≠ AC We must now show that the triangle satisfies IE = IF Sides of Triangle Side (c) Side AC = 2.00 ( ) AC is literally twice as long as AB! Bisector Segments Segment IE = 0.38 Segment IF = 0.38 Both segments are exactly identical!THE DIRECT VERDICT The proposition claims: If , then In our triangle: ✓ IE = IF = 0.38 (Hypothesis holds) × (Conclusion fails!) Therefore, the converse is FALSE. WHY ARE IE AND IF EQUAL? (3 SIMPLE STEPS) 1Angle between bisectors: In any triangle, the angle at incenter is: 2Opposite angles sum to 180°: Notice that . This guarantees quadrilateral is cyclic (its vertices lie on a single circle). 3Equal inscribed angles subtend equal chords: Line bisects , so . Chords facing equal inscribed angles inside the circle must be identical: This geometric proof relies only on . The sides and do not need to be equal at all!