Question

the angles of a right angled triangle are in arithmetic progression and the smallest side is 10.00m .the perimeter of the triangle in m is​

Answer 1

Answer:

Hence, the perimeter of triangle is:

10√3(√3+1) m. or in decimals we can say 47.32051 m.

Step-by-step explanation:

The angles of a right angled triangle are in arithmetic progression and the smallest side is 10 m.

Let the angles of a triangle be a, a+d and a+2d where d is the common difference.

Now we know that sum of all the angles of a triangle is 180°.

i.e.

a+a+d+a+2d=180°

3a+3d=180°

3(a+d)=180°

a+d=60° ( since on dividing both side by 3)

Now d=30

( since, a+2d=90

a+d=60

Hence d=30 )

Also,

a=30°

Hence, the three angles of a triangle are:

30°,60°,90°.

Now, as we know that the side opposite to the smallest angle is smallest.

Hence from the figure we get:

y=10 m.

$\tan 60=\dfrac{x}{y}\\\\\sqrt{3}=\dfrac{x}{y}\\\\x=\sqrt{3}y$

similarly,

$\sin 60=\dfrac{x}{h}\\\\\dfrac{\sqrt{3}}{2}=\dfrac{x}{h}\\\\h=\dfrac{2}{\sqrt{3}}x=\dfrac{2}{\sqrt{3}}\times \sqrt{3}h=2y$

Hence,

x=10√3 m.

h=20 m.

Hence, the perimeter of triangle is given as:

x+y+h

=10+10√3+20

=30+10√3

=10√3(√3+1) m.

Hence, the perimeter of triangle is:

10√3(√3+1) m. or in decimals we can say 47.32051 m.