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Higher Maths

Higher Maths. Get Started. Revision Notes. goodbye. Properties of the straight line You should know … ( x 1 , y 1 ) and x 2 , y 2 ) are points on a line …. ax + by + c = 0. recognise the term locus. the equation of a straight line given two points. y – y 1 = m ( x – x 1 ).

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Higher Maths

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  1. Higher Maths Get Started Revision Notes goodbye

  2. Properties of the straight line You should know … (x1, y1) and x2, y2) are points on a line … ax + by + c = 0 recognise the term locus the equation of a straight line given two points y–y1 = m(x–x1) the gradients of parallel lines are equal the equation of a straight line given one point and the gradient Lines are perpendicular if and only if m1m2 = –1 Find where lines intersect the concurrency properties of lines associated with the triangle

  3. y B(10,6) C A(–2,1) x O Using the formula … AB = √[(10 – (–2))2 + (6 – 1)2] = √(122 + 52) AB = √169 = 13 Thinking it out … AC = 10 – (–2) = 12 BC = 6 – 1 = 5 By Pythagoras’ theorem AB2 = 122 + 52 = 144 + 25 = 169 AB = √169 = 13 Test Yourself?

  4. y B(x2,y2) y B(6,12) Change in y … y2 – y1  x O  Change in x … x2 – x1 A(x1,y1)  x O  A(–1,–2) The gradient, m The gradient, m, is a measure of the slope of a line. It is defined to be The gradient Where ‘’ stands for ‘change in …’ We can see in the diagram that this is tan  and that  is also the angle the line makes with the x-axis. The angle Test Yourself?

  5. The equation of a line y Locusn, plloci a set of points whose position satisfies some condition or equation. The locus of the points(x, y) which satisfy the equation y = mx + c is a straight line with gradient m and y-intercept c. 3 2 y = 1/2x + 1 1 x 1 2 3 O y The equation of the line can take different forms • y = mx + c … gradient m and y-intercept c. • ax + by + c = 0 … a and b cannot both be zero. When a = 0 … we get an equation of the form y = k a line parallel to the x-axis When b = 0 … we get an equation of the form x = k a line parallel to the y-axis • y – y1 = m(x – x1) … where (x1, y1) lies on the line and m is its gradient. y = 3 3 2 x = –1 1 x – 4y + 2 = 0 x 1 2 3 O The line which passes through the points (–1, –5) and (1, 1) has a gradient Its equation is y – 1 = 3(x – 1) … or y – (–5) = 3(x – (–1)) Which both simplify to y = 3x – 2 The line which passes through the points (x1, y1) and (x2, y2) has a gradient and an equation y – y1 = m(x – x1) Test Yourself?

  6. Parallel and perpendicular To find the equation of the line parallel to y = 3x + 1 passing through the point (2, 4) When two lines are parallel their gradients are equal … m1 = m2 Parallel to y = 3x + 1 means its gradient is 3. Passing through (2, 4) means its equation is: y – 4 = 3(x – 2) When their gradients are equal, two lines are parallel … or the same line. When two lines are perpendicular The product of their gradients is negative 1. … m1m2 = – 1 To find the equation of the line perpendicular to y = 3x + 1 passing through the point (2, 4) Perpendicular to y = 3x + 1 means its gradient is –1/3. Passing through (2, 4) means its equation is: y – 4 = –1/3(x – 2) When the product of their gradients is negative 1, two lines are perpendicular Test Yourself?

  7. Intersecting lines When two lines intersect, find the point of intersection by solving their equations simultaneously. • Where do • y = 3x + 4 and y = 5x – 2 intersect? • 3x + 4 = 5x – 2 • 2x = 6 • x = 3 • y = 3.3 + 4 = 13 • intersect at (3, 13) If the equations are in the form y = mx + c … equate the expressions for y: m1x + c1 = m2x + c … solve for x … substitute and solve for y. • Where do • 2x + 3y – 9 =0 and 3x + 5y – 14 = 0 intersect? • 2x + 3y = 9 …  3x + 5y = 14 …   5: 10x + 15y = 45 …   3: 9x + 15y = 42 …  – : x = 3 • y = 1 (by substitution in ) • intersect at (3, 1) If the equations are in the form ax + by + c = 0 … use the elimination method … scaling the equations … subtract to eliminate one variable … solve … substitute and solve Test Yourself?

  8. B B B N M G A A M A C C C lines associated with the triangle B P A C M is the midpoint of BC. A is the vertex opposite BC. AM is called a median . M is the midpoint of BC. MN is perpendicular to BC. … a perpendicular bisector . G lies on BC. AG is perpendicular to BC. AG is an altitude . P lies on BC. AP bisects the angle at A AP is an angle bisector . More…

  9. B B B N M G A A M A C C C lines associated with the triangle B P A C A triangle has 3 medians. They are concurrent … [they all pass through the same point] The point of concurrency is called the centroid . A triangle has 3 perpendicular bisectors They are concurrent. The point of concurrency is called the circumcentre . A triangle has 3 altitudes They are concurrent. The point of concurrency is called the orthocentre . A triangle has 3 angle bisectors They are concurrent. The point of concurrency is called the incentre . More…

  10. B B N M A M A C C lines associated with the triangle B B N M P A A G C C The centroid cuts each median in the ratio 2:1 Its coordinates are: The circumcentre is the centre of the circumcircle … the circle which passes through the three vertices of the triangle The orthocentre, the centroid and the circumcentre all lie on the same line which is called the Euler line. The incentre is the centre of the incircle … the circle which has the three sides as tangents Test Yourself?

  11. reveal Between Central Station and The Kingston Bridge, Argyll St in Glasgow is a straight line. Relative to a suitable set of axes and using suitable units, the end-points are (–10, 30) and (73, 17). Calculate the length of Argyll St correct to the nearest unit.

  12. Between Central Station and The Kingston Bridge, Argyll St in Glasgow is a straight line. Relative to a suitable set of axes and using suitable units, the end-points are (–10, 30) and (73, 17). Calculate the length of Argyll St correct to the nearest unit. Argyll St = √[(–10 – 73)2 + (30 – 17)2] =√[(–83)2 + 132] = √(6889 + 169) = √7058 = 84 ( to nearest whole unit)

  13. y x O reveal A meteor streaks through the atmosphere. Relative to a suitable set of axes, it passes through the points (–1, 4) and (5, 1). What is the gradient of its flight path? At what angle does it cross the x-axis?

  14. y (a) x O A meteor streaks through the atmosphere. Relative to a suitable set of axes, it passes through the points (–1, 4) and (5, 1). What is the gradient of its flight path? At what angle does it cross the x-axis? (b) The negative sign tells us the angle is measured clockwise and is 53˚ below the x-axis 53˚ x

  15. reveal At an archaelogical dig a series of post holes seems to form a straight line. Relative to a convenient set of axes and units, there are holes at (–3, 5) and (6, 1). Find the equation of the line expressing it in the form ax + by + c = 0. Another post hole is discovered at (3, –2). Does this lie on the same line?

  16. (a) At an archaelogical dig a series of post holes seems to form a straight line. Relative to a convenient set of axes and units, there are holes at (–3, –5) and (6, 1). Find the equation of the line expressing it in the form ax + by + c = 0. Another post hole is discovered at (3, –2). Does this lie on the same line? • Consider the point (3, –2). • When x = 3 on the line • 2.3 – 3y – 9 = 0 • 3y = 6 – 9 • 3y = –3 • y = –1 • The point (3, –1) is on the line • (3, –2) is not. (b)

  17. y C B A O x reveal D A trapezium has vertices A(1, 0), B(3, 4), D(2, –3) and C. The coordinates of C are not given. AB is parallel to DC. Find the equation of both lines. When CB and DA are produced, they intersect at right angles. Find the equations of both lines.

  18. y C The equation of AB: B y – 0 = 2(x – 1)  y = 2x – 2 A O x The equation of DC: CD is parallel to AB so … D A trapezium has vertices A(1, 0), B(3, 4), D(2, –3) and C. The coordinates of C are not given. AB is parallel to DC. Find the equation of both lines. When CB and DA are produced, they intersect at right angles. Find the equations of both lines. y – (–3) = 2(x – 2)  y = 2x – 7 (b) The equation of DA: y – 0 = –3(x – 1)  y = –3x + 3 The equation of CB: CB is perpendicular to DA so …

  19. y C B A O x reveal D A trapezium has vertices A(1, 0), B(3, 4), D(2, –3) and C. The coordinates of C are not given. It can be found that the equation of DC is: 2x – y – 7 = 0 BC is: x –3y + 9 = 0 What are the coordinates of C?

  20. y C B A O x • 2x – y = 7 …  x – 3y = –9 …   3: 6x – 3y = 21 …   1: x – 3y = –9 …   –  : 5x = 30 • x = 6 • y = 5 (by substitution in ) • C is the point (6, 5) D A trapezium has vertices A(1, 0), B(3, 4), D(2, –3) and C. The coordinates of C are not given. It can be found that the equation of DC is: 2x – y – 7 = 0; of BC is: x –3y + 9 = 0 What are the coordinates of C?

  21. reveal Three businessmen living in Airdrie A(5, 0), Crieff C(11, 18) and Dunfermline D(13, 4) pick Stirling as a place to hold a meeting as it is equidistant from all three towns. This means that Stirling is the circumcentre of the triangle ACD. Find the coordinates of Stirling. [All coordinates are with reference to a convenient set of axes and units.]

  22. Perpendicular bisector of AD: Midpoint AD = (9, 2) So required equation is: y – 2 = –2(x – 9)  y = –2x + 20 Perpendicular bisector of DC: Three businessmen living in Airdrie A(5, 0), Crieff C(11, 18) and Dunfermline D(13, 4) pick Stirling as a place to hold a meeting as it is equidistant from all three towns. This means that Stirling is the circumcentre of the triangle ACD. Find the coordinates of Stirling. [All coordinates are with reference to a convenient set of axes and units.] Midpoint DC = (12, 11) So required equation is: y – 11 = 1/7(x – 12) Intersection of Perpendicular bisectors: • –2x + 20 – 11 = 1/7(x – 12) •  –14x + 140 – 77 = x – 12 • 15x = 75 • x = 5 • y = –2.5+ 20 • y = 10 • Stirling is at S(5, 10)

  23. This revision list takes you through the Learning outcomes required for Higher Maths, Unit 1, The Straight line. Practice makes Perfect.

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