4.  Which of the following can be the sides of a right triangle?

          (i) \small 2.5\hspace{1mm} cm, \small 6.5\hspace{1mm} cm, 6 cm.

          (ii) 2 cm, 2 cm, 5 cm.

         (iii) \small 1.5\hspace{1mm} cm, 2cm, \small 2.5\hspace{1mm} cm.

          In the case of right-angled triangles, identify the right angles.

Answers (1)

As we know, 

In a Right-angled Triangle: By Pythagoras Theorem,

(Hypotenus)^2=(Base)^2+(Perpendicular)^2

(i) \small 2.5\hspace{1mm} cm, \small 6.5\hspace{1mm} cm, 6 cm.

As we know the hypotenuse is the longest side of the triangle, So

Hypotenuse = 6.5 cm

Verifying the Pythagoras theorem,

 (6.5)^2=(6)^2+(2.5)^2

42.25=36+6.25

42.25=42.25

Hence it is a right-angled triangle.

The Right-angle lies on the opposite of the longest side (hypotenuse) So the right angle is at the place where 2.5 cm side and 6 cm side meet.

(ii) 2 cm, 2 cm, 5 cm.

As we know the hypotenuse is the longest side of the triangle, So

Hypotenuse = 5 cm

Verifying the Pythagoras theorem,

 (5)^2=(2)^2+(2)^2

25=4+4

25\neq8

Hence it is Not a right-angled triangle.

 (iii) \small 1.5\hspace{1mm} cm, 2cm, \small 2.5\hspace{1mm} cm.

As we know the hypotenuse is the longest side of the triangle, So

Hypotenuse = 2.5 cm

Verifying the Pythagoras theorem,

 (2.5)^2=(2)^2+(1.5)^2

6.25=4+2.25

6.25=6.25

Hence it is a Right-angled triangle.

The right angle is the point where the base and perpendicular meet.

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