Question

1. what is the ratio of the sides of triangle aby?

options:
1:2:√3
1:1:√2
1:2:√2
1:1:√3

Explanation:

Step1: Identify triangle type

Triangle ABY has a right angle at A, and $\angle ABY = 45^\circ$, so $\angle AYB = 180^\circ - 90^\circ - 45^\circ = 45^\circ$. This makes it an isosceles right triangle, meaning legs $AY = AB$.

Step2: Assign variable to legs

Let $AY = AB = x$.

Step3: Calculate hypotenuse YB

Use Pythagorean theorem: $YB = \sqrt{AY^2 + AB^2} = \sqrt{x^2 + x^2} = \sqrt{2x^2} = x\sqrt{2}$.

Step4: Find side ratio

The ratio of sides $AY:AB:YB = x:x:x\sqrt{2} = 1:1:\sqrt{2}$.

Answer:

1:1:$\sqrt{2}$ (Option B)