Question 1 - Think, Draw and Infer (Page 98) - Chapter 5 Class 9 - I’m Up and Down, and Round and Round (Ganita Manja

Transcript

Question 1 A,B and C are three collinear points. Can you find a point P such that PA=PB=PC? What can you say about the perpendicular bisectors of AB and BC? Draw and check. Can you show that for three collinear points A,B and C, the perpendicular bisector of AB and BC are parallel? Is it possible for a circle to pass through collinear points? Can you draw a line that cuts a given circle in three distinct points? Collinear points are those points which lie on the same line Let’s consider 3 points - A, B, C on the same line Now, let’s answer our questions one by one Can you find a point P such that PA=PB=PC? The question is asking if there is a point which is equal to all 3 points. That point would be the center of the circle passing through the 3 collinear points But, no circle passes through 3 collinear points. Thus, there is no point such that PA=PB=PC What can you say about the perpendicular bisectors of AB and BC? The perpendicular bisectors of AB and BC are parallel to each other Can you show that for three collinear points A,B and C, the perpendicular bisector of AB and BC are parallel? Points A, B, C are in the same line Now, Perpendicular bisector of AB is perpendicular to that line Perpendicular bisector of BC is perpendicular to that line Since both perpendicular bisectors are perpendicular to the same line. Thus, they are parallel Is it possible for a circle to pass through collinear points? To draw a circle through A, B, and C, you need a centre point (P) where the perpendicular bisectors intersect. Because the bisectors are parallel, they never intersect. No intersection = no centre point = no circle! Thus, it is not possible for circle to pass through collinear points Can you draw a line that cuts a given circle in three distinct points? A straight line can only cut a circle in a maximum of two points (creating a chord). Since a circle is curved, a straight line cannot bend to enter the circle, exit it, and enter it a third time. Thus, answer is no – a line cannot cut a circle in 3 distinct points