Sides of triangles are given below. Determine which of them are right triangles. In case of a right triangle, write the length of its hypotenuse.
(i),/b> 7 cm, 24 cm, 25 cm
(ii) 3 cm, 8 cm, 6 cm
(iii) 50 cm, 80 cm, 100 cm
(iv) 13 cm, 12 cm, 5 cm

To do:

We have to determine whether the triangle is a right-angled triangle and write the length of its hypotenuse in each case.

Solution:

(i) Let $a=7\ cm$, $b=24\ cm$ and $c=25\ cm$.

We know that,

If the square of the hypotenuse is equal to the sum of the squares of the other two sides, then the triangle is a right-angled triangle. 

Therefore,

$(a)^2=(7\ cm)^2=49\ cm^2$

$(b)^2=(24\ cm)^2=576\ cm^2$

$(c)^2=(25\ cm)^2=625\ cm^2$

Here, $(a)^2+(b)^2=(49+576)\ cm^2=625\ cm^2$

$(a)^2+(b)^2=(c)^2$

Therefore, by the converse of Pythagoras theorem, the given sides are the sides of a right triangle.

 The length of the hypotenuse is $25\ cm$.

(ii) Let $a=3\ cm$, $b=8\ cm$ and $c=6\ cm$.

We know that,

If the square of the hypotenuse is equal to the sum of the squares of the other two sides, then the triangle is a right-angled triangle. 

Therefore,

$(a)^2=(3\ cm)^2=9\ cm^2$

$(b)^2=(8\ cm)^2=64\ cm^2$

$(c)^2=(6\ cm)^2=36\ cm^2$

Here, $(a)^2+(c)^2=(9+36)\ cm^2=45\ cm^2$

$(a)^2+(c)^2≠(b)^2$

The square of larger side is not equal to the sum of squares of other two sides.

Therefore, the given triangle is not right angled.

(iii) Let $a=50\ cm$, $b=80\ cm$ and $c=100\ cm$.

We know that,

If the square of the hypotenuse is equal to the sum of the squares of the other two sides, then the triangle is a right-angled triangle. 

Therefore,

$(a)^2=(50\ cm)^2=2500\ cm^2$

$(b)^2=(80\ cm)^2=6400\ cm^2$

$(c)^2=(100\ cm)^2=10000\ cm^2$

Here, $(a)^2+(b)^2=(2500+6400)\ cm^2=8900\ cm^2$

$(a)^2+(b)^2≠(c)^2$

The square of larger side is not equal to the sum of squares of other two sides.

Therefore, the given triangle is not right angled.

(iv) $a=13\ cm$

$b=12\ cm$

$c=5\ cm$

We know that,

If the square of the hypotenuse is equal to the sum of the squares of the other two sides, then the triangle is a right-angled triangle. 

Therefore,

$(a)^2=(13\ cm)^2=169\ cm^2$

$(b)^2=(12\ cm)^2=144\ cm^2$

$(c)^2=(5\ cm)^2=25\ cm^2$

Here, $(b)^2+(c)^2=(144+25)\ cm^2=169\ cm^2$

$(b)^2+(c)^2=(a)^2$

Therefore, by the converse of Pythagoras theorem, the given sides are the sides of a right triangle.

 The length of the hypotenuse is $13\ cm$.