Question

Abc is a triangle right angled atbc and ac=√3 bc.Using cocept of geometrybprove that angle abc is 60 degree

Answer 1

Question : ABC is a triangle right angled at C, AC = $\sqrt{3}$ BC. Using the concept of Trigonometry, Prove that Angle ABC is 60°

Solution :

Triangle ABC is right angled at C, So C = 90°.

AB is the hyotenuse.

Now,

Consider tanB = AC /BC ===== ( 1)

From the question,

$AC = \sqrt{3} BC\\ \\ \frac{AC }{BC } = \sqrt{3}$

This is second equation, $\frac{AC }{BC } = \sqrt{3}$

From 1 & 2,

tan B = $\sqrt{3}$

tanB = tan60.

Therefore, Angle B or ABC = 60°

This is the method to solve using Trigonometry. If we are to do by Geometry, then the solution will be :

Let D be the midpoint of AB. Join CD.
Now, AB² = BC + AC²
AB² = BC² + (√3BC)²
AB² = BC² + 3BC² = 4BC²
AB² = 4BC²
AB = 2BC.
Now, BD = 1/2 AB = 1/2(2BC) = BC.
D is the midpoint of hypotenuse AB, Therefore it is equidistant from all the three vertices of ∆.
Therefore CD = BD = DA or CD = 1/2 AB = BC.
i.e., △BCD is a equilateral ∆.
Thus, BC = BD = CD
Hence, ∠ABC = 60°.