AreaAreas of Parallelograms and Triangles Maths - AreaAreas of Parallelograms and Triangles

Look at the statements given below:
$i.$ A parallelogram and a rectangle on the same base and between the same parallels are equal in area.
$ii.$ In a $\| gm \text{ABCD},$ it is given that $AB = 10\ cm.$
The altitudes $DE$ on $AB$ and $BF$ on $AD$ being $6\ cm$ and $8\ cm$ respectively, then $AD = 7.5\ cm.$
$iii.$ Area of a $\| gm =\frac{1}{2}\times\text{base}\times\text{altitude}.$
Which is true?

• A
  $I$ only
   
• B
  $II$ only
   
• ✓
  $I$ and $II$
   
• D
  $II$ and $III$

Answer: C.

In the given figure $ABCD$ is a trapezium in which $AB || DC$ such that $AB = a\ cm$ and $DC = b\ cm.$ If $E$ and $F$ are the midpoints of $AD$ and $BC$ respectively. Then, $ar(ABFE) : ar(EFCD) = ?$

• A
  $a : b$
• B
  $(a + 3b) : (3a + b)$
• ✓
  $(3a + b) : (a + 3b)$
• D
  $(2a + b) : (3a + b)$

Answer: C.

• ✓
  $1 : 2$
• B
  $1 : 3$
• C
  $1 : 4$
• D
  $3 : 4$

Answer: A.

In the given figure, $A B C D$ is a $\| g m$ in which diagonals $A C$ and $B D$ intersect at $O$. If ar $(\| g m A B C D)$ is $52 \mathrm{~cm}^2$, then the $\operatorname{ar}(\triangle \mathrm{AOB})=$ ?

• A
  $26 \mathrm{~cm}^2$
• B
  $18.5 \mathrm{~cm}^2$
• C
  $39 \mathrm{~cm}^2$
• ✓
  $13 \mathrm{~cm}^2$

Answer: D.

• A
  A median of a triangle divides it into two triangles of equal area.
• B
  The diagonals of a ||gm divide it into four triangles of equal area.
• C
  In a $\triangle\text{ABC},$ if $E$ is the midpoint of median $AD$, then $\text{ar}(\triangle\text{BED})=\frac{1}{4}\text{ar}(\triangle\text{ABC}).$
  
• ✓
  In a trapezium $ABCD,$ it is given that $AB || DC$ and the diagonals $AC$ and $BD$ intersect at $O.$ Then, $\text{ar}(\triangle\text{AOB})=\text{ar}(\triangle\text{ABC}).$
  
  

Answer: D.