Example 1 - Chapter 7 Class 10 Coordinate Geometry

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Example 1 Do the points (3, 2), ( 2, 3) and (2, 3) form a triangle? If so, name the type of triangle formed. Let the three points be P(3, 2), Q( 2, 3) & R(2, 3) We find the distances PQ, QR, and PR Calculating PQ x1 = 3 , y1 = 2 x2 = 2 , y2 = 3 PQ = (( 2 1)2+( 2 1)2) = (( 2 3)2+( 3 2)2) = (( 5)2+( 5)2) = ((5)2+(5)2) = ((5)2+(5)2) = (2(5)2) = 2 5 = 5 2 = 5 1.414 = 7.07 Calculating QR x1 = 2 , y1 = 3 x2 = 2 , y2 = 3 QR = (( 2 1)2+( 2 1)2) = (( 2 ( 2))2+(3 ( 3))2) = (( 2+2)2+(3+3)2) = (( 4)2+(6)2) = (( 4)2+(6)2) = (16+36) = 52 = 7.21 Calculating PR x1 = 3 , y1 = 2 x2 = 2 , y2 = 3 PR = (( 2 1)2+( 2 1)2) = (( 3 2)2+(2 3)2) = ((1)2+( 1)2) = (( 1)2+(1)2) = (1+1) = 2 = 1.141 Hence, PQ = 7.07 , QR = 7. 21, PR = 1.41 Since the sum of any two of these distances is greater than the third distance Therefore, P, Q, R form a triangle Also, PQ = 50 , QR = 52 , PR = 2 So, PQ2 + PR2 = ( 50)2 + ( 2)2 = 50 + 2 = 52 = (QR)2 Therefore, PQ2 + PR2 = QR2 So, PQR is a right angled triangle