Prove that in a triangle,other than an equila | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let the sides of the triangle be $a$,$b$,and $c$,and the angles opposite to them be $A$,$B$,and $C$ respectively.Assume $a$ is the longest side,so

Vedclass · Accepted Answer

Vedclass · Answer

(A) Let the sides of the triangle be $a$,$b$,and $c$,and the angles opposite to them be $A$,$B$,and $C$ respectively.
Assume $a$ is the longest side,so $a > b$ and $a > c$.
By the property of triangles,the angle opposite to the longest side is the largest angle,so $A > B$ and $A > C$.
We know that the sum of angles in a triangle is $A + B + C = 180^{\circ}$.
Since $A > B$ and $A > C$,we have $A + A + A > A + B + C$,which implies $3A > 180^{\circ}$.
Therefore,$A > 60^{\circ}$.
$A$ right angle is $90^{\circ}$,and $\frac{2}{3}$ of a right angle is $\frac{2}{3} \times 90^{\circ} = 60^{\circ}$.
Thus,$A > 60^{\circ}$ proves that the angle opposite to the longest side is greater than $\frac{2}{3}$ of a right angle.

Prove that in a triangle,other than an equilateral triangle,the angle opposite to the longest side is greater than $\frac{2}{3}$ of a right angle.

Similar Questions

In $\Delta ABC$,$\angle B = 50^{\circ}$ and $\angle C = 85^{\circ}$,then $AB$ $\dots$ $AC$.

In $\Delta PQR$,$PQ = PR$. If $\angle Q = 48^{\circ}$,then find $\angle P$ and $\angle R$.

In triangles $ABC$ and $DEF$,$\angle A = \angle D$,$\angle B = \angle E$ and $AB = EF$. Will the two triangles be congruent? Give reasons for your answer.

In the given figure,$PN$ and $QM$ are both perpendicular to line segment $PQ$. Also,$X$ is the midpoint of $PQ$ as well as $MN$. Prove that $\triangle PNX \cong \triangle QMX$.

Bisectors of the angles $B$ and $C$ of an isosceles triangle with $AB = AC$ intersect each other at $O$. $BO$ is produced to a point $M$. Prove that $\angle MOC = \angle ABC$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module