Question : For a triangle ABC, D and E are two points on AB and AC such that $\mathrm{AD}=\frac{1}{6} \mathrm{AB}$, $\mathrm{AE}=\frac{1}{6} \mathrm{AC}$. If BC = 22 cm, then DE is _______. (Consider up to two decimals)

Option 1: 1.33 cm

Option 2: 1.67 cm

Option 3: 3.67 cm

Option 4: 3.33 cm

Question : $\triangle$ABC is an equilateral triangle in which D, E, and F are the points on sides BC, AC, and AB, respectively, such that AD $\perp$ BC, BE $\perp$ AC and CF $\perp$ AB. Which of the following is true?

Option 1: 4AC$^2$ = 5BE$^2$

Option 2: 3AC$^2$ = 4BE$^2$

Option 3: 2AB$^2$ = 3AD$^2$

Option 4: 7AB$^2$ = 9AD$^2$

Question : ABC is an equilateral triangle points D, E and F are taken in sides AB, BC and CA respectively so that, AD = BE = CF. Then DE, EF, and FD enclose a triangle which is:

Option 1: equilateral

Option 2: isosceles

Option 3: right angled

Option 4: none

Question : $\triangle \mathrm{ABC}$ and $\triangle \mathrm{DEF}$ are two triangles such that $\triangle \mathrm{ABC} \cong \triangle \mathrm{FDE}$. If AB = 5 cm, $\angle$B = 40° and $\angle$A = 80°, then which of the following options is true?

Option 1: DF = 5 cm, $\angle$E = 60°

Option 2: DE = 5 cm, $\angle$F = 60°

Option 3:  DE = 5 cm, $\angle$D = 60°

Option 4: DE = 5 cm, $\angle$E = 60°

Question : In $\triangle{ABC}, \angle A=90^{\circ}, {AB}=16\ cm $ and $AC =12 \ cm$. $D$ is the midpoint of $AC$ and $DE \perp CB$ at $E$. What is the area (in ${cm}^2$ ) of $\triangle{CDE}$?

Option 1: 8.64

Option 2: 7.68

Option 3: 5.76

Option 4: 6.25