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Distance

C. A. B. D. y. x. Distance. Lengths parallel to the axes are calculated as if it was a number line. Lengths are always positive. Examples 1) Calculate the length of AB. 3 – –2 = 5 u. 2) Calculate the length of CD. 4 – –3 = 7 u. y. ( x 2 , y 2 ). y 2. y 1.

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Distance

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  1. C A B D y x Distance Lengths parallel to the axes are calculated as if it was a number line. Lengths are always positive. Examples 1) Calculate the length of AB. 3 – –2 = 5u 2) Calculate the length of CD. 4 – –3 = 7u

  2. y (x2, y2) y2 y1 (x1, y1) (x2, y1) x x1 x2 Distance To calculate the length of a sloping line we could: d (y2y1) • Draw a right angled triangle • Calculate the horizontal length (x2x1) (x2x1) • Calculate the vertical length (y2y1) • Which leads to this formula: • Use Pythagoras’ theorem If you forget this formula or forget how to apply it, just use Pythagoras’ theorem instead.

  3. Example 1 Calculate the distance between the points E(2, 5) and F(–4, 1). Does it matter which point is (x1, y1) or (x2, y2)? NO, either point can be!!! I would let the point with negative values be (x1, y1). If you need an exact answer, make sure you simplify any surds. If the question does not specify how it wants the answer, give an exact answer i.e. in surd form

  4. Example 2 C2 Book - Ex 4C (page 62) Q1 a, d, g, j, m, o Q 4, 6, 8 Then Q 5, 7, 9, 10 The coordinates of triangle BCD are B(8,2), C(11,13) and D(2,6) Prove that triangle is an isosceles, and give the length of the equal sides in units. Find the length of each side using Since there is a single pair of equal sides, the triangle BCD is isosceles, as required The equal side length is

  5. Corollary of Pythagoras Theorem To prove a right angle at ABC, show that AB2+BC2=AC2 C2 Book - Ex 4C (page 62) Q1 a, d, g, j, m, o Q 4, 6, 8 Then Q 5, 7, 9, 10 • 1 • 10 • d) • g) • j) 5c m) o)

  6. Hints Q8) If three points on the circumference of a circle make a right angle, what do we know about two points in particular (think Circle Theorems). Then use mid-point formula Q5) (a)Use mid-point formula (b) What would the distance from CP be equal to if P lies on the circle? Q7) Mid points and distance formula Q9) Corollary of Pythagoras. Then think about the formula for the area of a triangle. What do you need to know? Q10) Any parallelogram with two equal adjacent sides and one right angle is a square

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