Geometry Problem-Solving: Circles and Perpendicular Bisectors
Master the art of solving geometrical problems involving circles and perpendicular bisectors with step-by-step explanations. Learn how to find equations, gradients, and midpoints to crack challenging exercises.
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Teachings for Teachings for Exercise 6B Exercise 6B
Circles l Draw a sketch! The perpendicular bisector of a line segment ?? is the straight line that is perpendicular to ?? and passes through its midpoint (-1,4) A (5,2) B C The line AB is the diameter of the circle with centre C, where A and B are (-1, 4) and (5, 2) respectively. The line l passes through C and is perpendicular to AB. Find the equation of l. We need to: = -1/3 a) Find the gradient of the line AB b) Then work out the gradient perpendicular to that c) We also need to find the co- ordinates of the centre d) We can then find the equation of l y x y x = Gradient 2 1 2 1 = 3 2 4 5 ( 1) = Gradient 1 3 Gradient = 6B
Circles l Draw a sketch! The perpendicular bisector of a line segment ?? is the straight line that is perpendicular to ?? and passes through its midpoint (-1,4) A (5,2) B C The line AB is the diameter of the circle with centre C, where A and B are (-1, 4) and (5, 2) respectively. The line l passes through C and is perpendicular to AB. Find the equation of l. We need to: = -1/3 a) Find the gradient of the line AB b) Then work out the gradient perpendicular to that c) We also need to find the co- ordinates of the centre d) We can then find the equation of l + + ( x x y y Midpoint of AB = , 1 2 1 2 2 2 + = 3 + ( 1) 5 4 2 2 Midpoint of AB = , = (2,3) 2 ) Midpoint of AB = 2,3 6B
Circles l Draw a sketch! The perpendicular bisector of a line segment ?? is the straight line that is perpendicular to ?? and passes through its midpoint (-1,4) A (5,2) B C The line AB is the diameter of the circle with centre C, where A and B are (-1, 4) and (5, 2) respectively. The line l passes through C and is perpendicular to AB. Find the equation of l. We need to: = -1/3 a) Find the gradient of the line AB b) Then work out the gradient perpendicular to that c) We also need to find the co- ordinates of the centre d) We can then find the equation of l = = = = 2) ( ) y y y 3 m x 3( x x 1 1 = 3 3 3 6 y x = (2,3) 3 3 y x 6B
Circles (1,12) Draw a sketch! The perpendicular bisector of a line segment ?? is the straight line that is perpendicular to ?? and passes through its midpoint Q l C The line PQ is the Chord of the circle, centre (-3,5), where P and Q are (5,4) and (1,12) respectively. The line l is perpendicular to PQ and bisects it. Show that it passes through the centre of the circle. P (5,4) (-3,5) We need to: + + ( x x y y Midpoint of PQ = , 1 2 1 2 = (3,8) a) Find the midpoint of PQ b) Find the gradient of PQ, and then the perpendicular c) We can then find the equation of line l and substitute (-3,5) into it 2 2 + + 1 5 12 4 , 2 ) Midpoint of PQ = 2 Midpoint of PQ = 3,8 6B
Circles (1,12) Draw a sketch! The perpendicular bisector of a line segment ?? is the straight line that is perpendicular to ?? and passes through its midpoint Q l C The line PQ is the Chord of the circle, centre (-3,5), where P and Q are (5,4) and (1,12) respectively. The line l is perpendicular to PQ and bisects it. Show that it passes through the centre of the circle. P (5,4) (-3,5) We need to: y x y x = Gradient 2 1 = (3,8) a) Find the midpoint of PQ b) Find the gradient of PQ, and then the perpendicular c) We can then find the equation of line l and substitute (-3,5) into it 2 1 12 4 1 5 = -2 so 1/2 = Gradient Gradient = 2 6B
Circles (1,12) Draw a sketch! The perpendicular bisector of a line segment ?? is the straight line that is perpendicular to ?? and passes through its midpoint Q l C The line PQ is the Chord of the circle, centre (-3,5), where P and Q are (5,4) and (1,12) respectively. The line l is perpendicular to PQ and bisects it. Show that it passes through the centre of the circle. P (5,4) (-3,5) We need to: = ( ) y y m x 1 ( 2 0.5 x + x = + 0.5 6.5 y x 1 1 = (3,8) a) Find the midpoint of PQ b) Find the gradient of PQ, and then the perpendicular c) We can then find the equation of line l and substitute (-3,5) into it = 0.5( 3) 6.5 = + = 5 8 3) y x = -2 so 1/2 1.5 6.5 + 5 = = 8 0.5 1.5 y y x 6.5 6B
Circles y = 3x - 11 Draw a sketch! The perpendicular bisector of a line segment ?? is the straight line that is perpendicular to ?? and passes through its midpoint The lines AB and CD are chords of a circle. The line y = 3x 11 is the perpendicular bisector of AB. The line y = -x 1 is the perpendicular bisector of CD. Find the coordinates of the circle s centre. D C A THE PERPENDICULAR BISECTOR OF A CHORD GOES THROUGH THE CENTRE! B y = -x - 1 = = 3 4 11 11 1 x x x = x = x 1 We need to: 4 10 2.5 1) Set the bisectors equal to each other and solve the equation for x and y. y = 3.5 6B
