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:√2 √2:1:1 √3:2:1 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 ⇒1√2=xAC ⇒AC=x√2 ... (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: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. 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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