Question

A triangle and a parallelogram have the same base and the same area. If the sides of the triangle are 26cm, 28cm, 30cm and the parallelogram stands on the base 28cm, find the height of the parallelogram.

Answer 1

Hint: Let us use the formulas, area of triangle and area of parallelogram, and equate their areas as given in question. You will find the height of the parallelogram.

Formula used:

Area of triangle=  $\sqrt {s(s - a)(s - b)(s - c)} $ , where $s = \dfrac{{a + b + c}}{2}$; a, b, c are the three sides of the triangle. 

Area of parallelogram=  $b \times h$ , where ‘b’ is the base and ‘h’ is the height of the parallelogram.

Complete step-by-step solution:

Here we are given that the triangle and the parallelogram have the same area and same base and the sides of the triangle 26cm, 28cm, 30cm.

First, calculate the area of the triangle by substituting the values of the sides in the above mentioned area of the triangle formula.

Given a=26cm, b=28cm, c=30cm

$  s = \dfrac{{26 + 28 + 30}}{2} $

$  s = \dfrac{{84}}{2} $

$  s = 42 $  

Area of the triangle =  $\sqrt {s(s - a)(s - b)(s - c)} $  

= $\sqrt {42(42 - 26)(42 - 28)(42 - 30)} $ 

= $\sqrt {42 \times 16 \times 14 \times 12} $  

= $\sqrt {112896} $  

= $336c{m^2}$  $ \to eq(1)$ 

Area of the parallelogram = $b \times h$ 

$b = 28$  

$h = ?$  

Area of the parallelogram = $28 \times h \to eq(2)$  

Here, we know that the area of given triangle and parallelogram are same, so the equations 1 and 2 must be equated i.e. 

$\Rightarrow eq(1) = eq(2)$  

$\Rightarrow  336 = 28 \times h $

$\Rightarrow  h = \dfrac{{336}}{{28}} $

$  \therefore h = 12cm $ 

Therefore, the height of the given parallelogram with base 28cm is 12cm.

Note: Another solution to find the area of the triangle
Area of the triangle = $\dfrac{1}{2}(b \times h)$ , where ‘b’ is the base and ‘h’ is the height of the triangle
So, height is not given in the question which means we have to find the height of the triangle using the given three sides’ measurements.
$
  \dfrac{1}{2}(b \times h) = \sqrt {s(s - a)(s - b)(s - c)} \\
  h = \dfrac{{\sqrt {s(s - a)(s - b)(s - c)} \times 2}}{b} \\
$
Where a, b, c are the given three sides of the triangle and $s = \dfrac{{a + b + c}}{2}$ , where s is half the perimeter of the triangle.
In this way we can find the area of the triangle and using this we can find the height of the parallelogram according to the question.