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

The proofs for the side length ratios of 30-60-90 and 45-45-90 triangles are similar in using trigonometric ratios, but differ in specific ratios and angles used.

Step-by-step explanation:

The proofs for the side length ratios of 30-60-90 and 45-45-90 triangles are similar in that they both involve using trigonometric ratios to establish relationships between the side lengths. However, they are different in terms of the specific ratios used and the angles involved.

For the 30-60-90 triangle:

The side opposite the 30-degree angle is half the length of the hypotenuse.

The side opposite the 60-degree angle is the square root of 3 times the length of the side opposite the 30-degree angle.

For the 45-45-90 triangle:

Both legs are equal in length.

The length of the hypotenuse is the square root of 2 times the length of either leg.

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

`(AB)/(BC)=\tan 60^\circ\\\\\Rightarrow (AB)/(a)=\sqrt 3\\\\\Rightarrow AB = \sqrt 3~a.`

and

using Pythagoras theorem, we have

`AC=√(3a^2+a^2)=2a,`

`PR=√(b^2+b^2)=\sqrt 2~a.`

Therefore,

`(AB)/(PQ)=(\sqrt 3~a)/(b),~~~(BC)/(QR)=(a)/(b),~~~(AC)/(PR)=(2a)/(\sqrt 2~b).`

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