Question
A design is made on a rectangular tile of dimensions $50cm \times 70cm$ as shown in the design shows $8$ triangles, each of sides $26\ cm, 17\ cm$ and $25\ cm$. Find the total area of the design and the remaining area of the tile.
✓
Answer
Given, tha dimension of rectangular lile is $50 cm \times 70 cm$
Area of rectangular tile $=50 \times 70=3500 cm^2$
The sides of a design of one triangle be, $a=25 cm, b =17 cm$ and $c =26 cm$
Now, semi-perimeter, $\text{s}=\frac{\text{a}+\text{b}+\text{c}}{2}=\frac{25+17+26}{2}=\frac{68}{2}=34$
$\therefore$ Area of one triangle $=\sqrt{\text{s}(\text{s}-\text{a})(\text{s}-\text{b})(\text{s}-\text{c})}$ [by Heron's formula]
$=\sqrt{34\times9\times17\times8}$
$=\sqrt{17\times2\times3\times3\times17\times2\times2\times2}$
$=17\times3\times2\times2=204\text{cm}^2$
$\therefore$ Total area of eight triangles $=204 \times 8=1632 cm$
Now, area of the desion $=$ Total area of eight triangles $=1632 cm^2$
Also, remaining area of the tile
$=$ Area of the rectangle - Area of the design
$=3500-1632=1868 cm$
Hence, the total area of the design is $1632 cm^2$
and the remaining area of the tile is $1868 cm^2$
Area of rectangular tile $=50 \times 70=3500 cm^2$
The sides of a design of one triangle be, $a=25 cm, b =17 cm$ and $c =26 cm$
Now, semi-perimeter, $\text{s}=\frac{\text{a}+\text{b}+\text{c}}{2}=\frac{25+17+26}{2}=\frac{68}{2}=34$
$\therefore$ Area of one triangle $=\sqrt{\text{s}(\text{s}-\text{a})(\text{s}-\text{b})(\text{s}-\text{c})}$ [by Heron's formula]
$=\sqrt{34\times9\times17\times8}$
$=\sqrt{17\times2\times3\times3\times17\times2\times2\times2}$
$=17\times3\times2\times2=204\text{cm}^2$
$\therefore$ Total area of eight triangles $=204 \times 8=1632 cm$
Now, area of the desion $=$ Total area of eight triangles $=1632 cm^2$
Also, remaining area of the tile
$=$ Area of the rectangle - Area of the design
$=3500-1632=1868 cm$
Hence, the total area of the design is $1632 cm^2$
and the remaining area of the tile is $1868 cm^2$
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