The correct option is
A 1:1:√2 Let us assume the length of side AB of the triangle to be x units.
Applying trignometric ratios to the sides, we get :
sin 45∘=(xAC)⇒1√2=(xAC)⇒AC=x√2 ...(i)Similarly,tan 45∘=(xBC)⇒1=(xBC)⇒BC=x ...(ii) So, the ratios of the sides of the triangle with angles
45∘,45∘& 90∘=x:BC:AC =x:x:x√2(from (i)&(ii)) =1:1:√2 An alternate and shortcut method of solving this question is:
For the given triangle, as two angles are equal; the two sides opposite to these angles will also be equal.
And as the third angle is
90∘, the triangle is right - angled triangle.
Let us assume the length of the equal sides is equal to x units.
So, length of the hypotenuse =
√(x2+x2) =
x√2 So, Ratio of the sides of the triangle = x : x :
x√2 ⇒ Ratio of the sides of the triangle = 1 : 1 :
√2