Question

Prove that the points A (1, 7), B (4, 2), C (-1, -1) and D (-4, 4) are the vertices of a square.

Answer 1

Given :  A (1, 7), B (4, 2), C (-1, -1) and D (- 4, 4) .

 

To prove : points A (1, 7), B (4, 2), C (-1, -1) and D (-4, 4) are the vertices of a square.

 Solution :  

By using distance formula : √(x2 - x1)² + (y2 - y1)²

Vertices : A (1, 7), B (4, 2)

Length of side AB = √(4 - 1)² + (2 - 7)²

AB = √3² + (-5)²

AB = √9 + 25

AB = √34 units

 

Vertices : B (4, 2), C (-1, -1)

Length of side BC = √(- 1 - 4)² + ( - 2 - 1)²

BC = √(-5)² + (-3)²

BC = √25 + 9

BC = √34 units

 

Vertices : C (-1, -1) and D (-4, 4)

Length of side CD = √(- 4 + 1)² + (4 + 1)²

CD = √(-3)² + (5)²

CD = √9 + 25

CD = √34 units

 

Vertices : D (-4, 4) , A (1, 7)

Length of side DA = √(- 4 - 1)² + (4 - 7)²

DA = √(-5)² + 3²

DA = √25 + 9

DA = √34 units

 

Vertices : B (4, 2), D (-4, 4)

Length of diagonal BD = √(- 4 - 4)² + (4 - 2)²

BD = √(-8)² + 2²

BD = √64 + 4

BD = √68 units

 

Vertices : A (1, 7) , C (-1, -1)

Length of diagonal AC = √(- 1 - 1)² + (-1 - 7)²

AC = √(-2)² + (-8)²

AC = √4 + 64

AC = √68 units

Since all the four sides (AB = BC = CD = DA = √34)  and diagonal  (BD = AC = √68) both are equal.

Hence, the given vertices are the vertices of a square.

Some more questions :  

Show that the quadrilateral whose vertices are (2, −1), (3, 4) (−2, 3) and (−3,−2) is a rhombus.

https://brainly.in/question/15937739

 

Prove that the points (−2, 5), (0, 1) and (2, −3) are collinear.

https://brainly.in/question/15937742

Answer 2

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(a) Draw a line segment AC=6 cm.

(b) At point C, draw XC perpendicular CA.

(c) Taking C as centre and radius 6 cm, draw an arc.

(d) This arc cuts CX at point B.

(e) Join BA.

It is the required isosceles right angled triangle ABC