Distance between two points
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The document discusses two approaches to calculating the distance between two points: the traditional distance formula approach and an alternative approach using a drawing and the Pythagorean theorem. It provides an example of finding the distance between points (-3,4) and (5,-3) and step-by-step works through the alternative approach of drawing a right triangle between the points and applying the Pythagorean theorem to calculate the distance as the hypotenuse of 10.6 units. The document suggests asking students to estimate distances, draw diagrams, calculate, and reflect on their work.
Distance between two points
- 1. Topic: the distance between two points. TRADITIONAL APPROACH: Distance-formula: y Q (x2,y2) P (x ,y ) 1 1 0 x The distance between P and Q ( PQ ) is (x 2 −1 )2 +y 2 − 1 )2 x ( y Example: Find the distance between the points with coordinates (-3,4) and (5,-3). Solution: ( (−3) − 5) 2 + ( 4 − (−3) ) 2 = 64 + 49 = 113 ≈ 10.6 ALTERNATIVE APPROACH: Start: Find the distance between the points with coordinates (-3,4) and (5,-3). STEP 1: Draw the situation! (The situation will get more meaning)
- 2. y (-3,4) 1 0 1 x (5,-3) STEP 2: So, what do you think the distance between the two points will be? 3? Or 5? Or greater ? Why? (ESTIMATE!!) STEP 3: We are going to find out. We will use a right triangle, and Pythagorean theorem. y C (-3,4) 1 0 1 x B A (5,-3) Recall:Pythagorean Theorem for the right triangle above: AB2+AC2=BC2 • What is the distance between A and B? (You can just count: 8) (USE OF COMMON SENSE, OF WHAT YOU ALREADY KNOW) (If you want to express it already in terms of the x-coordinates: | 5-(-3)|=8; but maybe better to first show that you don’t need any ‘complex’ notations…)
- 3. • What is the distance between A and C? (You can just count: 7) (USE OF COMMON SENSE, OF WHAT YOU ALREADY KNOW) (If you want to express it already in terms of the y-coordinates: | (-3)-4|=7; but maybe better to first show that you don’t need any ‘complex’ notations…) • So what is the distance between B and C? With use of Pythagorean theorem: (BC)2= 82+72 = 64+49 = 113, so the distance between B and C is 113 = 10.6 (CALCULATE!!) STEP 4: From your first estimation and the drawing, you think 10.6 could be the correct answer? So you did not make any mistakes? (REFLECTION!!) After this, you can give one other example, or you can already let them work. You could even ask the students to give a general formula of the distance between two points, if they have practised enough exercises. You can also ask a CONTEXT-problem in which distance is concerned, for instance:
- 4. Mark lives 3 km from school and Rona lives 5 km from school. a) What is the maximum possible distance between their two houses and what is the minimum possible distance between their two houses? b) If the situation is as illustrated below, how far do Mark and Rona live apart? Rona school Mark c) If you know that Rona and Mark live 4 km apart, draw in the graph below the possible locations for Mark’s house: Rona school How did you find the locations?