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Question

If a triangle has angles 45,45, and 90, what is the ratio of the sides of the triangle opposite to these angles respectively?


  1. 1:1:2

  2. 2:1:1

  3. 3:2:1

  4. 1:2:3


Solution

The correct option is A

1:1:2




Let us assume the length of side AB of the triangle to be x.
Applying trignometric ratios to the sides, we get :
sin45=xAC
12=xAC
AC=x2 ... (i)
Similarly,
tan45=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:x2  [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.
So, length of the hypotenuse =x2+x2=2x  [Pythagoras theorem]
So, Ratio of the sides of the triangle =x:x:2x
 Ratio of the sides of the triangle =1:1:2

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