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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 $13\ cm$.