Krishna
0

Step 1: Write the ratios in the form of lengths by using any variable.

NOTE: We know that there is a common ratio for all sides of the triangle.

That means that if that common ratio is denoted as x

EXAMPLE:  Lengths a = 3x

b = 4x

c = 6x

Step 2: Use the Law of Cosines to find the large angle

Step 1:  Recall the "Law of Cosines"

NOTE:ab and c are sides.

C is the angle opposite side c

The Law of Cosines (also called the Cosine Rule) says:

c^2 = a^2 + b^2 - 2ab cos (C)

\cos\left(C\right)=\frac{a^2+b^2\ -\ c^2}{2ab}

Step 2: Substitute the given measurements in the Cosine rule

NOTE: We could have taken each of the ratio results in another

order, of course. Now, what matters is that the largest angle in the

triangle is opposite to the longest side

EXAMPLE: \cos\ C\ =\ \frac{\left(3x\right)^2\ +\ \left(4x\right)^2\ -\ \left(6x\right)^2}{2\cdot3x\cdot4x}

Step 3: Do some calculations

EXAMPLE: \cos\ C\ =\frac{\ 11x^2}{-24\ x^2}\ =\ -\frac{11}{24}

Cos C = - 0 . 45833

Step 4: Send the Cos to R.H.S side.

NOTE: Find the Cos inverse value

EXAMPLE: C = \cos^{-1}\ \left(-\ 0.4833\right)

Step 3: Conclude that the triangle type based on the angle.

NOTE:

c^2 = a^2 + b^2, then the angle at C

C is a right angle.

c^2 < a^2 + b^2, then the angle at C

C is acute.

c^2 > a^2 + b^2,  then the angle at C

C is obtuse.

The cosine of an obtuse angle is always negative,

so it follows that the triangle (having this type of angle) is obtuse.