Choose whether the given statement is true or false.

Two triangles having the same base (or equal base) and equal areas lie between the same parallels.

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True

False

answer is A.

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Detailed Solution

Two triangles ABC and ABD are located on the same base AB, so Area (ABC) = Area (ABD) are located between the same parallels. We can draw the altitudes of ABC and ABCD through C and D, respectively, and meet the bases AB at E and F.
Question Image

Now, we have to prove CD⊥AB.
Since CE and DF are perpendicular to AB, CE and DFare parallel to each other.
Since lines perpendicular to the same line are parallel to each other.
∴CE⊥DF
Now, we are given that Area (ΔABC) = Area (ΔABD)
Area (ΔABC) =

\frac{1}{2}

×b × h =

\frac{1}{2}

× AB × CE
Area (ΔABD) =

\frac{1}{2}

× b × h =

\frac{1}{2}

× AB × DF
Since, Area (ΔABC) = Area(ΔABD)
⇒

\frac{1}{2}

×AB×CE=

\frac{1}{2}

×AB×DF
⇒CE=DF
Now, in CDFE, we have
⇒CE=DF and CE⊥DF
Because two opposite sides are parallel and equal, CDFE is a parallelogram.
We know that parallelograms' opposite sides are parallel.
As a result, CDEF
Because EF is on the plane AB, we have CDAB. As a result, proved.
Therefore, two triangles having the same base (or equal base) and equal areas lie between the same parallels.
So, the given statement is true.

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