Question

The area of the triangle is equal to the area of the rectangle whose length and breadth are \[20{\text{ }}cm\] and \[15{\text{ }}cm\] respectively. Calculate the height of the triangle if its base measures\[30{\text{ }}cm\].

Answer 1

Hint: The formula for the area of a rectangle whose length and breadth are ‘\[l\]’ and ‘\[b\]’ respectively is;
Area of rectangle $ = l \times b $
The formula for the area of a triangle whose height and base are ‘h’ and ‘b’ respectively is;
Area of triangle $ = \dfrac{1}{2} \times b \times h $

Complete step-by-step answer:
We are given the dimensions of the rectangle in the question, that is;
Length ( $ l $ ) of the rectangle $ = 20\;cm $
Breadth ( $ b $ ) of the rectangle $ = 15\;cm $
We are also given the base length of the triangle;
Base ( $ s $ ) $ = 30\;cm $
Now from the dimensions of the rectangle we can find its area since both its length and breadth are given. So the area of rectangle is as follows;
 $ \Rightarrow $ Area of rectangle ( $ Ar $ ) $ = l \times b $
 $ \Rightarrow $ Area of rectangle ( $ Ar $ ) $ = 20 \times 15\;c{m^2} $
 $ \Rightarrow $ Area of rectangle ( $ Ar $ ) $ = 300\;c{m^2} $
But in the question it is already given that the area of the rectangle is actually exactly equal to the area of the triangle, so the area of the triangle will also be
 $ \Rightarrow $ Area of triangle ( $ At $ ) $ = 300\;c{m^2} $
Now we know that the formula of the area of the triangle is $ \dfrac{1}{2} \times s \times h $ , where \[s\]is the base length and \[h\]is the height of the triangle.
From the question we are given that the length of the base of this triangle is $ 30\;cm $ , and now that we have the value of the area of the triangle that is $ 300\;c{m^2} $ , we shall substitute these values into the formula of the triangle. This will be as follows;
 $ \Rightarrow $ Area of triangle ( $ At $ ) $ = \dfrac{1}{2} \times b \times h $
 $ \Rightarrow $ $ 300 $ $ = \dfrac{1}{2} \times 30 \times h $
 $ \Rightarrow $ $ 300 $ $ = 15 \times h $
 $ \Rightarrow $ $ \dfrac{{300}}{{15}} = h $
 $ \therefore h = 20\;cm $
So therefore the required height of the triangle is $ 20\;cm $ .
So, the correct answer is “ $ 20\;cm $ ”.

Note: Sometimes the three sides of a triangle might be mentioned in the question. If that is the case then we can use Heron's formula to calculate the area of the triangle with the three given sides. First we can calculate the perimeter, take its half and use it in Heron’s formula along with the three individual sides as shown below:
Area $ = \sqrt {s(s - a)(s - b)(s - c)} $ ;
where $ a,b,c $ are the sides of the triangle and $ s = \dfrac{{(a + b + c)}}{2} $