Straight Lines

Coordinate geometry joins algebra and geometry by describing geometric figures with equations. In earlier classes you used the distance formula, the section formula and the area of a triangle in the coordinate plane. This chapter studies the simplest curve, the straight line. It begins with the inclination of a line and its slope m = tan(theta) = (y2 - y1)/(x2 - x1). From this you get the conditions for two lines to be parallel (equal slopes) or perpendicular (product of slopes equal to -1), the test for collinearity of three points, and the angle between two lines from tan(theta) = (m2 - m1)/(1 + m1 m2), taken as a positive value for the acute angle. You will then learn the various forms of the equation of a line: lines parallel to the axes, the point-slope form, the two-point form, the slope-intercept form y = mx + c and the intercept form x/a + y/b = 1. The chapter ends with a formula for the perpendicular distance of a point from a line.

Detailed Worksheet: Straight Lines Section A - Definitions (10 marks) 1. Find the slope of the line through (3, -2) and (-1, 4). (2 marks) 2. Write the slope of a line whose inclination is 60 degrees. (2 marks) 3. Find x if the slope of the line through (2, 5) and (x, 3) is 2. (2 marks) 4. Write the equation of the line parallel to the x-axis passing through (2, -3). (2 marks) 5. State the conditions for two non-vertical lines to be parallel and perpendicular. (2 marks) Section B - Calculations and Applications (15 marks) 6. Show that the points (1, -1), (2, 1) and (4, 5) are collinear. (3 marks) 7. Find the equation of the line through (-2, 3) with slope -4. (3 marks) 8. Find the equation of the line through (1, -1) and (3, 5). (3 marks) 9. Find the acute angle between two lines whose slopes are 1/2 and 3. (3 marks) 10. Find the slope and the y-intercept of 3x + 4y = 12. (3 marks) Section C - Diagrams (10 marks) 11. Draw the line 2x + 3y = 6 by finding its intercepts on the axes. (4 marks) 12. Draw a diagram showing the inclination theta of a line and explain why its slope is tan(theta). (3 marks) 13. Draw a diagram showing the perpendicular distance of a point P from a line L. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Derive the point-slope form and the two-point form of the equation of a line. (5 marks) 15. Find the equation of the line perpendicular to x - 7y + 5 = 0 that has x-intercept 3. (5 marks) 16. Find the equation of the line through (-3, 5) perpendicular to the line through (2, 5) and (-3, 6). (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Write final equations in the form ax + by + c = 0. mod(x) denotes the absolute value of x.