Question

A(8,2) and B(6,4) are the vertices of a figure which is symmetrical about x = 6 and y = 2. Complete the figure and give the geometrical name of the figure. 

Answer 1

Steps of construction: 

(i) Plot the point A and B on the graph. 

(ii) Plot point P whose vertices are x = 6 and y = 2. Pis the point of symmetry. 

(iii) Point symmetric to A(8,2) in the line x = 6 is C( 4,2) 

(iv) Point symmetric to B(6,4) in the line y = 2 is D(6,0) 

(v) Join AP, PC, BP and PD. 

Since BD = 4,

AD = `sqrt ((8 - 6)^2 + (2 - 0)^2) = sqrt (2^2 + 2^2) = sqrt (4 + 4) = sqrt 8`

AB = `sqrt ((8 - 6)^2 + (2 - 4)^2) = sqrt (2^2 + (-2)^2) = sqrt (4 + 4) = sqrt 8`

∴ BD² = AD² + AB²

`4^2 = (sqrt 8)^2 + (sqrt 8)^2 `

16 =  8 + 8

16 = 16

∴ ∠ BAD = 90°

Clearly AB=BC=CD=DA ,∠ BAD = 90° and AC and BD bisect each other at right angles 1 therefore ABCD is a square.