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  • A design is made on a rectangular tile of dimensions 50 cm × 70 cm as shown in Figure. 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.

A design is made on a rectangular tile of dimensions 50 cm × 70 cm as shown in Figure. 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.

                                                    

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[The\; area\; of\; the\; design\; is\; 1632\; cm^{2}\; and\; the\; remaining \; area\; of\; the \; tile \; is\; 1868\; cm^{2}.]

We have dimensions of rectangle tile as 50 cm × 70 cm

We know that area of rectangle = length × breadth

Area \; of\; tile = (70 × 50)\; cm^{2} = 3500\; cm^{2}

Given sides of triangular design: 26 cm, 17 cm, 25 cm

To find the area using Heron’s formula

Let, a = 26 cm, b = 17 cm, c = 25 cm

S=\frac{a+b+c}{2}=\frac{26+17+25}{2}=\frac{68}{2}=34\; cm

Area\; of\; triangle =\sqrt{S\left ( S-a \right )\left ( S-b \right )\left ( S-c \right )}

=\sqrt{34\left ( 34-26 \right )\left ( 34-17 \right )\left ( 34-25 \right )}

= \sqrt{34\times 8\times 17\times 9}

= \sqrt{17\times 2\times 2\times 2\times 2\times 17\times 3\times 3}

= 2\times 2\times 3\times 17

Area of \DeltaABC = 204 cm^{2}

But we have 8 triangles of equal area

So area of design = 8 × area of one \Delta

                            = 8 × 204 = 1632 cm^{2}

Remaining area of tile = Area of tile - Area of design

= (3500 – 1632) cm^{2} = 1868 cm^{2}

Hence the area of the design is 1632 cm^{2} and the remaining area of the tile is 1868 cm^{2}.

 

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