Coordinate Geometry – Outcomes Use slopes to show that lines are either parallel or perpendicular. Find the intersection point of two lines. Solve problems about slopes of lines. Solve problems about the perpendicular distance from a point to a line. Solve problems about the angle between two lines. Divide a line segment internally in a given ratio a:b. Calculate the area of a triangle.

Use Slopes to Show that Lines are Perpendicular or Parallel Parallel lines have the same slope. i.e. 𝑚 1 = 𝑚 2 Perpendicular lines have inverse, negative slopes. i.e. 𝑚 1 × 𝑚 2 =−1⇒ 𝑚 2 =− 1 𝑚 1 When a line is expressed in the form 𝑦=𝑚𝑥+𝑐, its slope can simply be read from the equation as the coefficient of 𝑥. When a line is in the form 𝑎𝑥+𝑏𝑦=𝑐, its slope is − 𝑎 𝑏 .

Use Slopes to Show that Lines are Perpendicular or Parallel e.g. given lines 𝑙:𝑦=4𝑥−7 and 𝑘:8𝑥+𝑐𝑦=9 Find the slope of 𝑙 Find the slope of 𝑘 in terms of 𝑐. Find the value of 𝑐 if 𝑙∥𝑘 (i.e. 𝑙 is parallel to 𝑘). Find the value of 𝑐 if 𝑙⊥𝑘. (i.e. 𝑙 is perpendicular to 𝑘).

Find the Intersection Point of Two Lines By plotting lines on a coordinate plane, their intersection point can be found. Find the intersection point of these lines by drawing a graph: 𝑥+𝑦=4; 𝑥−𝑦=10 𝑥+𝑦=6; 2𝑥−4𝑦=12 5𝑥−2𝑦=9; 3𝑥+𝑦=1

⊥ Distance from a Point to a Line For a line 𝑎𝑥+𝑏𝑦+𝑐=0 and a point ( 𝑥 1 , 𝑦 1 ), the perpendicular distance between them (also the shortest path) is given by the formula: 𝑝= 𝑎 𝑥 1 +𝑏 𝑦 1 +𝑐 𝑎 2 + 𝑏 2 Note the modulus which ensures a positive result.

⊥ Distance from a Point to a Line e.g. Find the perpendicular distance from the point (4,−1) to the line 3𝑥−4𝑦=7 ⇒3𝑥−4𝑦−7=0 = 𝑎 𝑥 1 +𝑏 𝑦 1 +𝑐 𝑎 2 + 𝑏 2 = 3×4−4×−1−7 3 2 + −4 2 = 12+4−7 9+16 = |9| 5 = 9 5

⊥ Distance from a Point to a Line e.g. Find the equations of the two lines which contain the point (-5, -4) and which are at a distance 17 from the point (1, 3).

⊥ Distance from a Point to a Line The co-ordinates of three points A, B, and C are: A(2, 2), B(6, – 6), C(–2, –3). Find the perpendicular distance from C to AB.

Angle Between Two Lines For two lines with slopes 𝑚 1 and 𝑚 2 , the angle between the lines is given by: tan 𝜃 =± 𝑚 1 − 𝑚 2 1+ 𝑚 1 𝑚 2 Whether you get the acute or obtuse angle between the lines depends on the signs of 𝑚 1 and 𝑚 2 , and whether you take the + or − from the ± sign.

Angle Between Two Lines e.g. Two lines have equations 2𝑥+𝑦−5=0 and 𝑥+5𝑦− 2=0 respectively. Find the obtuse angle between the two lines. e.g. Find the equations of the two lines through the point (1, -2) which make angles of tan −1 1 2 with the line 4𝑥−𝑦+ 5=0.

Angle Between Two Lines Three points A, B and C have co-ordinates: A(-2, 9), B(6, -6), and C(11, 6). The line l passes through B and has equation 12𝑥− 5𝑦−102=0. Verify that C lines on l. Find the slope of AB and hence find tan (∠𝐴𝐵𝐶) as a fraction.

Divide a Line Segment Given a line segment [AB], with A( 𝑥 1 , 𝑦 1 ) and B( 𝑥 2 , 𝑦 2 ), we can divide it in the ratio a:b by finding the point C along the line: 𝐶= 𝑏 𝑥 1 +𝑎 𝑥 2 𝑏+𝑎 , 𝑏 𝑦 1 +𝑎 𝑦 2 𝑏+𝑎 e.g. If A(3, -1) and B(-9, 1), find the coordinates of C which divides [AB] in the ratio 3:1 𝐶= 1×3+3×−9 3+1 , 1×−1+3×1 3+1 𝐶= 3−27 4 , −1+3 4 𝐶=(−6, 0.5)

Divide a Line Segment If a = (2, 1) and b = (-8, 6), find the coordinates of the point c , if c divides [ab] internally in the ratio 3 : 2. Dividing [AB] in the ratio 5:2 yields the point C(-5, -4). If A(-5, 1), find the coordinates of B(x, y). A(8, 3) and C(-1, 2) are two points. Find B if it divides [AC] internally in the ratio 3:1 Given 𝑓 is a transformation which moves points: 𝑥, 𝑦 →( 𝑥 ′ , 𝑦 ′ ), where 𝑥 ′ =2𝑥−𝑦 and 𝑦 ′ =𝑥+2𝑦, find f(A), f(B), and f(C). Verify that f(B) divides [f(A)f(C)] in the ratio 3:1

Calculate the Area of a Triangle For a triangle OAB with one point, O, at the origin, 𝐴= ( 𝑥 1 , 𝑦 1 ) and 𝐵=( 𝑥 2 , 𝑦 2 ), the area of the triangle is given by: 𝐴𝑟𝑒𝑎= 1 2 | 𝑥 1 𝑦 2 − 𝑥 2 𝑦 1 | Note the modulus signs, making sure the answer is always positive (since area cannot be negative).

Calculate the Area of a Triangle e.g. What is the area of triangle OAB if 𝑂=(0, 0), 𝐴= (1, 7), and 𝐵=(5, −2)? 𝐴𝑟𝑒𝑎= 1 2 1 −2 − 7 5 = 1 2 −2−35 = 1 2 −37 = 37 2

Calculate the Area of a Triangle If none of the vertices are on the origin, the triangle must be translated. e.g. Find the area of triangle ABC if 𝐴=(3, −1), 𝐵=(4, 2), and 𝐶=(−1, 3). Choose one point to translate to (0, 0) and apply that translation to each point: 𝐴 3, −1 −3, +1 𝐴 ′ 0, 0 𝐵 4, 2 −3, +1 𝐵 ′ 1, 3 𝐶(−1, 3) −3, +1 𝐶′(−4, 4)

Calculate the Area of a Triangle 𝐴 ′ =(0, 0), 𝐵 ′ =(1, 3), 𝐶 ′ =(−4, 4) 𝐴𝑟𝑒𝑎= 1 2 𝑥 1 𝑦 2 − 𝑥 2 𝑦 1 = 1 2 1 4 − 3 −4 = 1 2 4+12 = 1 2 16 =8

Area of Triangles The line RS cuts the x-axis at the point R and the y-axis at the point S (0, 10), as shown. The area of the triangle ROS, where O is the origin, is 125 3 . Find the coordinates of R. E(-5, 4) is on the line RS. A second line 𝑦=𝑚𝑥+𝑐 passes through E and also makes a triangle of area 125 3 with the axes. Find the values of 𝑚 and 𝑐 if they are both positive.