Question

If a triangle has angles ${45}^{\circ },{45}^{\circ }$, and ${90}^{\circ }$, what is the ratio of the sides of the triangle opposite to these angles respectively?

• A. $1:1:\sqrt{2}$
• B. $\sqrt{2}:1:1$
• C. $\sqrt{3}:2:1$
• D. $1:2:\sqrt{3}$

Answer 1

The correct option is A

$1:1:\sqrt{2}$

Let us assume the length of side $AB$ of the triangle to be $x$.
Applying trignometric ratios to the sides, we get :
$\sin {45}^{\circ }=\frac{x}{AC}$
$\Rightarrow \frac{1}{\sqrt{2}}=\frac{x}{AC}$
$\Rightarrow AC=x\sqrt{2}$ ... (i)
Similarly,
$\tan {45}^{\circ }=\frac{x}{BC}$
$\Rightarrow 1=\frac{x}{BC}$
$\Rightarrow BC=x$ ... (ii)
So, the ratios of the sides of the triangle with angles ${45}^{\circ },{45}^{\circ }\&{90}^{\circ }=x:BC:AC$
$=x:x:x\sqrt{2}$ [from (i) & (ii)]
$=1:1:\sqrt{2}$

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 ${90}^{\circ }$, the triangle is right - angled triangle.
Let us assume the length of the equal sides is equal to $x$.
So, length of the hypotenuse $=\sqrt{x^2+x^2}=\sqrt{2}x$ [Pythagoras theorem]
So, Ratio of the sides of the triangle $=x:x:\sqrt{2}x$
$\Rightarrow$ Ratio of the sides of the triangle $=1:1:\sqrt{2}$