Question 9 - End-of-Chapter Exercises - Chapter 1 Class 9 - Orienting Yourself: The Use of Coordinates (Ganita

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Question 9 The following table shows the coordinates of points S, M and T. In each case, state whether M is the midpoint of segment ST. Justify your answer When M is the mid-point of ST, can you find any connection between the coordinates of M, S and T? For mid-point, we can do this by two ways Logically By Formula Formula for finding mid-point is Mid-point of (x1, y1) & (x2, y2) = ((𝒙_𝟏 + 𝒙_𝟐)/𝟐,(𝒚_𝟏+𝒚_𝟐)/𝟐) We do it by both methods Row 1: 𝑺(−𝟑,𝟎),𝑴(𝟎,𝟎),𝑻(𝟑,𝟎). Logically Journey 𝐒→𝐌 : You start at 𝑥=−3 and walk to 𝑥=0. That is 𝟑 steps right. Journey 𝐌→𝐓 : You start at 𝑥=0 and walk to 𝑥=3. That is also 3 steps right. Logical Answer: Yes, the steps are identical, so 𝑀 is exactly in the middle. Formula Here, S (–3, 0) and T (3, 0) Mid-point of S & T = ((−3 + 3)/2,(0 + 0)/2) = (0/2,0/2) = (0, 0) This is coordinates of point M So, point M is in the middle Row 2: 𝑺(𝟐,𝟑),𝑴(𝟑,𝟒),𝑻(𝟒,𝟓) Logically Journey 𝐒→𝐌 : 𝑥 goes from 2 to 3 (+1 step). 𝑦 goes from 3 to 4 (+1 step) Journey 𝐌→𝐓 : 𝑥 goes from 3 to 4 (+1 step). 𝑦 goes from 4 to 5 (+1 step). Logical Answer: Yes, you took the exact same diagonal path both times, so 𝑀 is perfectly halfway. Formula Here, S (2, 3) and T (4, 5) Mid-point of S & T = ((2 + 4)/2,(3 + 5)/2) = (6/2,8/2) = (3, 4) This is coordinates of point M So, point M is in the middle Row 3: 𝑺(𝟎,𝟎),𝑴(𝟎,𝟓),𝑻(𝟎,−𝟏𝟎). Logically Journey 𝐒→𝐌:𝑦 goes from 0 up to 5 ( 𝟓 steps up). Journey 𝐌→𝐓 : 𝑦 goes from 5 all the way down to -10 (15 steps down). Logical Answer: No, moving 5 steps is not the same as moving 15 steps. 𝑀 is way closer to 𝑆. Formula Here, S (0, 0) and T (0, –10) Mid-point of S & T = ((0 + 0)/2,(0 +(−10))/2) = (0/2,(−10)/2) = (0, –5) This is not the coordinates of point M So, point M is not in the middle Row 4: S (-8, 7), 𝑴(𝟎,−𝟐),𝑻(𝟔,−𝟑) Logically Journey 𝐒→𝐌:𝑥 goes from -8 to 0 (+8 steps horizontally). Journey 𝐌→𝐓 : 𝑥 goes from 0 to 6 (+6 steps horizontally). Logical Answer: No. Since the horizontal steps don't even match ( 8 vs 6 ), 𝑀 cannot be in the middle, so we don't even need to check the y -coordinates. Formula Here, S (–8, 7) and T (6, –3) Mid-point of S & T = ((−8 + 6)/2,(7 + (−3))/2) = ((−(8 − 6) )/2,(7 − 3)/2) = ((−2)/2,4/2) = (–1, 2) This is not the coordinates of point M So, point M is not in the middle The Connection (Midpoint Formula): To find the midpoint, you simply average the x coordinates and average the y –coordinates of points S (x1, y1) and T (x2, y2) Coordinates of M = ((𝒙_𝟏 + 𝒙_𝟐)/𝟐,(𝒚_𝟏+𝒚_𝟐)/𝟐)