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Question
Find the value of k, if the points A(7, −2), B (5, 1) and C (3, 2k) are collinear.
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Solution
The formula for the area ‘A’ encompassed by three points(x1,y1) , (x2 , y2) and (x3 , y3) is given by the formula,
\[∆ = \frac{1}{2}\left| \left( x_1 y_2 + x_2 y_3 + x_3 y_1 \right) - \left( x_2 y_1 + x_3 y_2 + x_1 y_3 \right) \right|\]
If three points are collinear the area encompassed by them is equal to 0.
The three given points are A(7, −2), B(5, 1) and C(3, 2k). It is also said that they are collinear and hence the area enclosed by them should be 0.
\[∆ = \frac{1}{2}\left| \left( 7 \times 1 + 5 \times 2k + 3 \times - 2 \right) - \left( 5 \times - 2 + 3 \times 1 + 7 \times 2k \right) \right|\]
\[ 0 = \frac{1}{2}\left| \left( 7 + 10k - 6 \right) - \left( - 10 + 3 + 14k \right) \right|\]
\[ 0 = \frac{1}{2}\left| - 4k + 8 \right|\]
\[ 0 = - 4k + 8\]
\[ k = 2\]
Hence the value of ‘k’ for which the given points are collinear is k = 2 .
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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
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OR
What is the area of Trapezium AFGH?
