Question

# The sides of a triangular plate are $8cm$,$15cm$and$17cm$.If its weight is $96gm$,find the weightof the plate per square $cm$.$\left( A \right)$.$1.6gm$\left( B \right).0.8gm$\left( C \right)$.$1.4gm$$\left( D \right)$.$2gm$

Hint: Use Heronâ€™s formula to compute the area of the triangular plate and find weight per square cm by dividing the weight by area of the triangle.

Let $ABC$be the triangular plate with sides
$AB = 17cm \\ AC = 15cm \\ BC = 8cm \\$
Given the problem, the weight of this triangular plate is $w = 96gm$.
In order to find the weight of the plate per square $cm$, we first need to compute the area of the
same.
Since sides of the triangular plate are given, we can use Heronâ€™s formula to calculate the area of the
plate.
Heronâ€™s formula states that area of the triangle is given by,
$\Delta = \sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} {\text{ (1)}}$
In the above equation, $s$ is the semi-perimeter of the triangle and $a,b,c$ are the sides of the
triangle.
Sides of the triangular plate $ABC$are given by
$\left( a = AB = 17cm \\ b = BC = 8cm \\ c = AC = 15cm \\ \right){\text{ (2)}}$
Semi-perimeter of triangle $ABC$ is given by,
$s = \dfrac{{a + b + c}}{2} \\ \Rightarrow s = \dfrac{{17 + 8 + 15}}{2} = 20cm{\text{ (3)}} \\$
Using equation $(2)$and $(3)$ in $(1)$,we get
$\Delta = \sqrt {20\left( {20 - 17} \right)\left( {20 - 8} \right)\left( {20 - 15} \right)} \\ \Rightarrow \Delta = \sqrt {20\left( 3 \right)\left( {12} \right)\left( 5 \right)} = 60c{m^2} \\$
Therefore, weight of the triangular plate per square $cm = \dfrac{w}{\Delta } = \dfrac{{96gm}}{{60c{m^2}}} = 1.6gm$per $c{m^2}$.
Hence the correct option is $\left( A \right)$.$1.6gm$ .

Note: Heronâ€™s formula should be used to compute the area where sides of the triangle are given. In
the problems like above, units need to be mentioned in the final answer.