A design is made on a rectangular tile of dimensions `"50 cm" xx "17"` `cm` as shown in figure. The design shows `8` triangle, each of sides `26 cm, 17 cm` and `25 cm.` Find the total area of the design and the remaining area of the tiles.

Explanation

Given, the dimensions of rectangular tile are `50 m` `xx` `70 cm.`
`:.` Area of rectangular tile `=50xx70`
`= 3500 cm^(2)`
The sides of a design of one triangle be
`a = 25 cm , b = 17 cm` and `c = 26 cm.`
Now, semi -perimeter, `s=(a+b+c)/(2)`
`=(5+17+26)/(2)=(68)/(2)=34` cm
`:.` Area of `DeltaABC=sqrt(s(s-a)(s-b)(s-c)) " "` [by Heron's formula]
`=sqrt(34xx9xx17xx8)`
`sqrt(17xx2xx3xx3xx17xx2xx2xx2)`
`=17xx3xx2xx2`
`=204 cm^(2)`
`:. ` Total area of eight triangles `=204xx8=1632 cm^(2)`
Now, area of the design= total area of eight triangles
`=1632 cm^(2)`
Also, remaining area of the tile= area of the rectangle - area of the design
`=3500 cm^(2) - 1632 cm^(2)`
` = 1868 cm^(2)`
Hence, the total area of the design is `1632 cm^(2)` and the remaining area of the tile is 1868 `cm^(2)`.