Question

Angles of a right triangle are in a arithmetic sequence. a) find the middle term of the sequence? b) write the angles of the triangles?

Answer 1

Answer

The middle term of the AP is 60°

The angles of the triangle are 90° , 60° and 30°

$\bf\large\underline{Given}$

• Angles of a right triangle are in arithmetic sequence.

$\bf\large\underline{To \: Find}$

• The middle term of the sequence
• The angles of the triangle

$\bf\large\underline{Solution}$

Let us consider the angles of the triangle be a - d , a , a + d (since they are in AP)

Thus , from the properties of triangle we have ,

$\sf\implies Sum \: of \: the \: angles = 180\degree \\\\ \sf\implies a-d + a + a + d = 180\degree \\\\ \sf\implies 3a = 180\degree \\\\ \sf\implies a = \dfrac{180\degree}{3} \\\\ \sf\implies a = 60\degree$

Thus , middle term of the AP is 60°

Again, it is given that the triangle is a right angled triangle

Therefore , its largest angle is 90°

$\sf\implies a + d = 90\degree \\\\ \sf\implies 60\degree + d = 90\degree \\\\ \sf\implies d = 90\degree - 60\degree \\\\ \sf\implies d = 30\degree$

Thus , common difference of the AP is 30°

The remaining angle of the triangle is :

$\sf\longrightarrow a - d \\\\ \longrightarrow\sf 60\degree -30\degree \\\\ \sf\longrightarrow 30\degree$

Therefore ,the angles of the triangle are 90° , 60° and 30°