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Last updated at May 29, 2018 by Teachoo

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Example 2 Show that the points (1, 7), (4, 2), ( 1, 1) and ( 4, 4) are the vertices of a square. Let the points be A( 1, 2) , B(1, 0) , C( 1, 2) , D( 3, 0) To prove that ABCD is a square, We have to prove all sides equal, i.e. AB = BC = CD = AD & diagonals equal i.e. AC = BD We find the distances AB, BC, CD & AD and AC & BD Finding AB x1 = 1 , y1 = 7 x2 = 4 , y2 = 2 AB = (( 2 1)2+( 2 1)2) = ((4 1)2+(2 7)2) = ((3)2+( 5)2) = ((3)2+(5)2) = (9+25) = 34 Finding BC x1 = 4 , y1 = 2 x2 = 1 , y2 = 1 BC = (( 2 1)2+( 2 1)2) = (( 1 4)2+( 1 2)2) = (( 5)2+( 3)2) = ((5)2+(3)2) = (25+9) = 34 Finding CD x1 = 1 , y1 = 1 x2 = 4 , y2 = 4 CD = (( 2 1)2+( 2 1)2) = (( 4 ( 1))2+(4 ( 1))2) = (( 4+1)2+(4+1)2) = (( 3)2+(5)2) = ((3)2+(5)2) = (9+ 25) = 34 Finding AD x1 = 1 , y1 = 7 x2 = 4 , y2 = 4 AD = (( 2 1)2+( 2 1)2) = (( 4 1)2+(4 7)2) = (( 5)2+( 3)2) = ((5)2+(3)2) = (25+9) = 34 Now, finding diagonals Finding AC x1 = 1 , y1 = 7 x2 = 1 , y2 = 1 AC = (( 2 1)2+( 2 1)2) = (( 1 1)2+( 1 7)2) = (( 2)2+( 8)2) = ((2)2+(8)2) = (4+64) = 68 Now, finding diagonals Finding BD x1 = 4 , y1 = 2 x2 = 4 , y2 = 4 BD = (( 2 1)2+( 2 1)2) = (( 4 4)2+(4 2)2) = (( 8)2+(2)2) = ((8)2+(2)2) = (64 +4) = 68 Since, AB = BC = CD = AD = 34 All sides of ABCD are equal, & AC = BD = 68 Hence, both diagonals are equal Since, all sides and both diagonals are equal Hence, ABCD is a square

Class 10

Important Questions for Exam - Class 10

- Chapter 1 Class 10 Real Numbers
- Chapter 2 Class 10 Polynomials
- Chapter 3 Class 10 Pair of Linear Equations in Two Variables
- Chapter 4 Class 10 Quadratic Equations
- Chapter 5 Class 10 Arithmetic Progressions
- Chapter 6 Class 10 Triangles
- Chapter 7 Class 10 Coordinate Geometry
- Chapter 8 Class 10 Introduction to Trignometry
- Chapter 9 Class 10 Some Applications of Trignometry
- Chapter 10 Class 10 Circles
- Chapter 11 Class 10 Constructions
- Chapter 12 Class 10 Areas related to Circles
- Chapter 13 Class 10 Surface Areas and Volumes
- Chapter 14 Class 10 Statistics
- Chapter 15 Class 10 Probability

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.