Question

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

Answer 1

Solution:

Consider the Triangle ABC in which , ∠A=60°, ∠B=90°,∠C=30°

Let , Perpendicular =  P

We will use trigonometric ratio to prove that ratio of sides  will always be same by using either of angles that is 30° or 60°.

Sin C= $\frac{\text{Perpendicular}}{\text{Hypotenuse}}$

Sin 30°= $\frac{1}{2}$

So, Ratio of Perpendicular and Hypotenuse is always constant, that is always  $\frac{1}{2}$.

Therefore , Hypotenuse will be = 2 P

Now, using Pythagoras Theorem

⇒(Perpendicular)² + (Base)² = (Hypotenuse)²

⇒P²+ (Base)²=(2 P)²

⇒ (Base)²= 4 P²- P²

⇒(Base)²= 3 P²

⇒ Base= √3 P

So, ratio of Perpendicular to Base or Base to Perpendicular is $\frac{1}{\sqrt{3}} or \frac{\sqrt{3}}{1}$.

Sin 60°= $\frac{BC}{AC}=\frac{\sqrt3P}{P}=\frac{\sqrt3}{1}$

⇒If you consider either angle of 30° or 60° to find the side length ratios, , then also ratio of sides is always that is  BC: AB or BC: AC or AB: AC →→ AB : BC : AC = 1 : √3 : 2 .

now, consider Triangle P QR, in which ∠P=∠R= 45°,∠Q=90°

To find the side length ratio we will use trigonometric ratio.

→Tan 45°= $\frac{PQ}{QR}$

→1= $\frac{PQ}{QR}$

→P Q= QR

Let , P Q= QR= K

Using Pythagoras theorem

⇒(Perpendicular)² + (Base)² = (Hypotenuse)²

⇒ K² + K²= (Hypotenuse)²

⇒(Hypotenuse)²= 2 K²

Hypotenuse= √2 K

⇒If you consider either angle that is of 45°  to find the side length ratios, , then also ratio of sides is always that is  P Q: QR or P Q: PR or QR: PR →→ P Q : QR : PR = 1 : 1:√2.