Draw: Figures with equal areas

Triangle Equal in Area to Given Quadrilateral

Construct a triangle equal in area to given quadrilateral

Construction:

1. $ABCD$ is a given quadrilateral.

   

2. Join $A$ to $C$.

   

3. Through $D$, draw $\overline{DE}$ parallel to $\overline{AC}$ meeting $\overline{BC}$ produced at point $E$.

   

4. Join $A$ to $E$, then $ABE$ is the required triangle.

   

Rectangle Equal in Area to Given Triangle

Construction:

1. Draw a triangle $ABC$.

   

2. Draw a perpendicular bisector ${PD}$ of $\overline{BC}$.

   

3. Through $A$, draw a line $PQ$ parallel to $\overline{BC}$.

   

4. Take $m\overline{PQ}=m\overline{DC}$.

   

5. Then $CDPQ$ is the required rectangle.

   

Square Equal in Area to Given Rectangle

Construction:

1. $ABCD$ is a given rectangle.

   

2. Produce side $\overline{AD}$ to $E$ making $m\overline{DE} = m\overline{CD}$.

   

3. Bisect $\overline{AE}$ at $M$.

   

4. With centre $M$ and radius $m\overline{AM}$ construct a semicircle.

   

5. Produce $\overline{CD}$ to meet the semicircle at $F$.

   

6. On $\overline{DF}$ as a side construct a square $DFQP$. This shall be required square.

   

Triangle of Equivalent Area on a Base of Given Triangle

Construction:

1. $ABC$ is given triangle.

   

2. Draw $AD \parallel \overline{BC}$.

   

3. With $B$ as centre, and radius = x, such that $m\overline{BC}=x$ draw an arc cutting $AD$ at P.

   

4. Join $\overline{BP}$ and $\overline{CP}$.

   

5. Then $\triangle BCP$ is the required triangle with equal base $\overline{BP} = x$ and area equivalent to $\triangle ABC$.