Question

A tiling or tessellation of a flat surface is the covering of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. Historically, tessellations were used in ancient Rome and in Islamic art. You may find tessellation patterns on floors, walls, paintings etc. Shown below is a tiled floor in the archaeological Museum of Seville, made using squares, triangles and hexagons.

A craftsman thought of making a floor pattern after being inspired by the above design. To ensure accuracy in his work, he made the pattern on the Cartesian plane. He used regular octagons, squares and triangles for his floor tessellation pattern

Use the above figure to answer the questions that follow:

1. What is the length of the line segment joining points B and F?
2. The centre ‘Z’ of the figure will be the point of intersection of the diagonals of quadrilateral WXOP. Then what are the coordinates of Z?
3. What are the coordinates of the point on y-axis equidistant from A and G?
   OR
   What is the area of Trapezium AFGH?

Answer 1

i. B(1, 2), F(–2,9)

BF² = (–2 –1)² + (9 – 2)²

= (–3)² + (7)²

= 9 + 49

= 58

So, BF = `sqrt(58)` units

ii.

W(–6, 2), X(–4, 0), O(5, 9), P(3, 11)

Clearly, WXOP is a rectangle

The point of intersection of the diagonals of a rectangle is the midpoint of the diagonals. So the required point is the mid point of WO or XP.

= `((-6 + 5)/2, (2 + 9)/2)`

= `((-1)/2, 11/2)`

iii. A(–2, 2), G(–4, 7)

Let the point on y-axis be Z(0, y)

AZ² = GZ²

(0 + 2)² + (y – 2)² = (0 + 4)² + (y – 7)²

( 2)² + y² + 4 – 4y = (4)² + y² + 49 – 14y

8 – 4y = 65 – 14y

10y = 57

So, y = 5.7

i.e. the required point is (0, 5.7)

OR

A(–2, 2), F(–2, 9), G(–4, 7), H(–4, 4)

Clearly GH = 7 – 4 = 3 units

AF = 9 – 2 = 7 units

So, height of the trapezium AFGH = 2 units

So, area of AFGH = `1/2`(AF + GH) × height

= `1/2`(7 + 3) × 2

= 10 sq.units