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 : In a triangle ABC, AD, BE, and CF are three medians. The perimeter of ABC is:

Option 1: equal to $(\overline {AD}+\overline{BE}+\overline{CF})$

Option 2: greater than $(\overline {AD}+\overline{BE}+\overline{CF})$

Option 3: less than $(\overline {AD}+\overline{BE}+\overline{CF})$

Option 4: none of these

Question : In a $\triangle ABC$, D and E are two points on AB and AC respectively such that DE || BC, DE bisects the $\triangle ABC$ in two equal areas. Then the ratio DB : AB is:

Option 1: $1:\sqrt2$

Option 2: $1:2$

Option 3: $\left ( \sqrt2-1 \right ):\sqrt2$

Option 4: $\sqrt2:1$

Question : The sides AB, BC, and AC of a $\triangle {ABC}$ are 12 cm, 8 cm, and 10 cm respectively. A circle is inscribed in the triangle touching AB, BC, and AC at D, E, and F respectively. The difference between the lengths of AD and CE is:

Option 1: 4 cm

Option 2: 5 cm

Option 3: 3 cm

Option 4: 2 cm

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°