Question

State whether the given statement is true or false.
Two triangles on the same base (or equal bases) and between the same parallels are equal in area.

Answer 1

Hint: Try to make a diagram to solve the question and use the formula of area of the triangle which is equal to half of the product of the base and its height, that is
Area of a triangle\[ = \dfrac{1}{2} \times Base \times Height\].

Complete step-by-step answer:
It is given that there are two triangles on the same base and between the same parallel lines.
We have to show that the area of both the triangles are the same.
Let’s make a line AD which is parallel to a line BC, Draw two lines are drawn namely AB and DC perpendicular to the baseline BC and join the point A and D with the midpoint of line BC which is E. now, two triangles are made which are $\Delta ABE$ and $\Delta DCE$with the same bases which is line BC.

As bases are same in both the triangles $\Delta ABE$ and $\Delta DCE$, then we have
$BE = CE$(Given)
It is given that both the triangles lie between the same parallel lines so we can conclude that:
$AB = DC$ … (1)
(As any line which is drawn and intersects two parallel lines will always be of same length, just as concrete which intersects railway tracks).
Now, we will find the area of a Triangle \[ = \dfrac{1}{2} \times Base \times Height\]\[\]
Now, substitute the values of $\Delta ABE$ and $\Delta DCE$ in the formula of the area of a triangle.
$Area\left( {\Delta ABE} \right) = \dfrac{1}{2}\left( {AB} \right)\left( {BE} \right)$
$Area\left( {\Delta DCE} \right) = \dfrac{1}{2}\left( {DC} \right)\left( {CE} \right)$ … (2)
Putting the value of the equation (1) into (2), then
\[Area\left( {\Delta DCE} \right) = \dfrac{1}{2}\left( {AB} \right)\left( {BE} \right)\]
Equating the areas of triangle $ABC$ & $DBC$,
$Area\left( {\Delta ABE} \right) = Area\left( {\Delta DCE} \right)$
Hence proved that two triangles which are on the same base and are between the same parallel lines are equal in area.

Note: We can also solve it using the congruence. We can easily show that two sides and one angle in both the triangles are same so using the SAS congruence both the triangles are congruent and thus the area of both the triangles are same.