Areas of Parallelograms and Triangles Maths - Areas 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 $\text{ABCD}$ 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 $, \text{ar(ABFE) : ar(EFCD)} =$ ?

• 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, $\text{ABCD}$ is a $\| g m$ in which diagonals $\text{AC}$ and $\text{BD}$ intersect at $O$. If ar $(\| g m \operatorname{ABCD})$ is $52 \ cm^2$, then the $\operatorname{ar}(\triangle AOB )=?$

• A
  $26\ cm^2$
   
• B
  $18.5\ cm^2$
   
• C
  $39\ cm^2$
  
   
• ✓
  $13\ cm^2$

Answer: D.

Which of the following is a false statement?

• 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 $\text{ABCD},$ 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.