Question : In a triangle, $ABC, A P$, the bisector of $\angle A$, is perpendicular to $B C$ at point $P$. The measures of $BP$ and $PC$ are $x$ and $3y$, respectively. The measures of $AB$ and $AC$ are $4 x$ and $(5 y+21)$, what is the value of $(x+y)$?

Option 1: 21

Option 2: 15

Option 3: 12

Option 4: 18

Question : In $\triangle$ABC, BD and CE are perpendicular to AC and AB respectively. If BD = CE, then $\triangle$ABC is:

Option 1: Equilateral

Option 2: Isosceles

Option 3: Right–angled

Option 4: Scalene

Question : In $\triangle ABC$, $D$ and $E$ are the points of sides $AB$ and $BC$ respectively such that $DE \parallel AC$ and $AD : DB = 3 : 2$. The ratio of the area of trapezium $ACED$ to that of $\triangle DBE$ is:

Option 1: $4:15$

Option 2: $15:4$

Option 3: $4:21$

Option 4: $21:4$

Question : In $\triangle ABC, \angle B = 60^\circ$ and $\angle C = 40^\circ$, AD and AE are respectively the bisectors of $\angle A$ and perpendicular on BC. Find the measure of $\angle EAD$.

Option 1: $11^\circ$

Option 2: $10^\circ$

Option 3: $12^\circ$

Option 4: $9^\circ$

Question : Three angles of a triangle are $(x-15^{\circ}),(x+45^{\circ}),$ and $(x+60^{\circ})$. Identify the type of triangle.

Option 1: Obtuse angle triangle

Option 2: Right angle triangle

Option 3: Isosceles triangle

Option 4: Equilateral triangle

Question : If $\frac{xy}{x+y}=a$, $\frac{xz}{x+z}=b$ and $\frac{yz}{y+z}=c$, where $a,b,c$ are all non-zero numbers, $x$ equals to:

Option 1: $\frac{2abc}{ab+bc–ac}$

Option 2: $\frac{2abc}{ab+ac–bc}$

Option 3: $\frac{2abc}{ac+bc–ab}$

Option 4: $\frac{2abc}{ab+bc+ac}$