Question

How are the proofs for the side length ratios of 30-60-90 and 45-45-90 triangles similar? How are they different?

Answer 1

Answer:

The two triangles are not similar

Step-by-step explanation:

Given side length ratios of 30-60-90 and 45-45-90 triangles. we have to tell are these two triangles similar.

To prove the two triangles similar either the angles are similar or the sides are in same proportion.

In ΔABC, the sides are in ratio $AB:BC:AC=\sqrt3:1:2$ and

In ΔPQR, the sides are in ratio $PQ:QR:PR=1:1:\sqrt2$

Hence, the sides

$\frac{AB}{PQ}={\sqrt3}{1}\\\\\frac{BC}{QR}={1}{1}\\\\\frac{AC}{PR}={2}{\sqrt2}$

The sides are not in same proportion.

⇒ No similarity rule apply

Hence, These two triangles are not similar

Answer 2

Answer:  The triangles are not similar.

Step-by-step explanation:  As given in the question and shown in the attached figure, ΔABC and ΔPQR are drawn, where

∠A = 30°, ∠B = ∠Q = 90°, ∠C = 60° and ∠P = ∠R = 45°.

Let us assume that BC = a units and PQ = QR = b units.

Then

$\dfrac{AB}{BC}=\tan 60^\circ\\\\\Rightarrow \dfrac{AB}{a}=\sqrt 3\\\\\Rightarrow AB = \sqrt 3~a.$

and

using Pythagoras theorem, we have

$AC=\sqrt{3a^2+a^2}=2a,$

$PR=\sqrt{b^2+b^2}=\sqrt 2~a.$

Therefore,

$\dfrac{AB}{PQ}=\dfrac{\sqrt 3~a}{b},~~~\dfrac{BC}{QR}=\dfrac{a}{b},~~~\dfrac{AC}{PR}=\dfrac{2a}{\sqrt 2~b}.$

Since the ratios are not equal, so the triangles are not similar.