Question
Determine whether the triangle having sides $(a - 1) cm, 2\sqrt{\text{a}}\text{ cm}$ and $(a + 1)\ cm$ is a right angled triangle.
✓
Answer
Sides of a triangle are $(a - 1)\ cm, 2\sqrt{\text{a}}\text{ cm}$ and $(a + 1)\ cm$
Let,$AB = (a - 1)cm, BC = (a + 1)cm$
and $\text{AC}=2\sqrt{\text{a}}$
$ \text { Now } A B^2=(a-1)^2=a^2-2 a+1 $
$ B C^2=(a+1)^2=a^2+2 a+1 $
$ A C^2=(2 \sqrt{a})^2=4 a $
$ \text { Now } A B^2+A C^2=a^2-2 a+1+4 a $
$ =a^2+2 a+1 $
$ =B C^2$
$\therefore\triangle\text{ABC}$ is a right triangle right angle at $\angle\text{A}$
(By converse of pythagoras theorem)
Let,$AB = (a - 1)cm, BC = (a + 1)cm$
and $\text{AC}=2\sqrt{\text{a}}$
$ \text { Now } A B^2=(a-1)^2=a^2-2 a+1 $
$ B C^2=(a+1)^2=a^2+2 a+1 $
$ A C^2=(2 \sqrt{a})^2=4 a $
$ \text { Now } A B^2+A C^2=a^2-2 a+1+4 a $
$ =a^2+2 a+1 $
$ =B C^2$
$\therefore\triangle\text{ABC}$ is a right triangle right angle at $\angle\text{A}$
(By converse of pythagoras theorem)
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