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# There is a slide in a children park. The front side of the slide has been painted and a message “ ONLY FOR CHILDREN” is written on it. If the sides of the triangular front wall of the slide are 8m , 9m and 3m, then find the area which is painted in colour.A. $2\sqrt {35} {m^2}$B. $4\sqrt {35} {m^2}$C. $2\sqrt {31} {m^2}$D. $3\sqrt {35} {m^2}$

Last updated date: 12th Aug 2024
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Hint: We are given that front side of the slide is painted and the front side is the shape of a triangle and with the given sides we can find the area of the triangular front wall using heron’s formula, Area of the triangle = $\sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)}$ , where $s = \dfrac{{a + b + c}}{2}$ and a , b , c are the given sides of the triangle.

Complete step by step solution:
We are given the diagram of the slide

And we can see that the front wall is in the shape of a triangle and that is the area which is painted
And we are given the sides of the triangle to be 8m, 9m and 3 m
Since we are given only the sides we need to use the heron’s formula to find the area of the triangle
Area of the triangle = $\sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)}$ , where $s = \dfrac{{a + b + c}}{2}$
Here a = 9 m , b = 8 m and c = 3 m
Using this we get
$\Rightarrow s = \dfrac{{9 + 8 + 3}}{2} \\ \Rightarrow s = \dfrac{{20}}{2} = 10m \\$
Using this in the heron’s formula we get the area to be
$\Rightarrow Area = \sqrt {10\left( {10 - 9} \right)\left( {10 - 8} \right)\left( {10 - 3} \right)} \\ \Rightarrow Area = \sqrt {10\left( 1 \right)\left( 2 \right)\left( 7 \right)} \\ \Rightarrow Area = \sqrt {\left( {10} \right)\left( {14} \right)} \\ \Rightarrow Area = \sqrt {\left( {140} \right)} = \sqrt {4\text 35} \\ \Rightarrow Area = 2\sqrt {35} {m^2} \\$
From this we get the area of the wall which is painted is $2\sqrt {35} {m^2}$

Therefore the correct answer is option A.

Note:
1) Here we don’t use the regular area formula as we are not given the height of the triangle.
2) The area can be defined as the space occupied by a flat shape or the surface of an object.
3) The area of a figure is the number of unit squares that cover the surface of a closed figure. 4) Area is measured in square units such as square centimeters, square feet, square inches, etc.