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

$\;\;\underline{\textbf{\textsf{ Given:-}}}$

• Angles of a right triangle

( In arithmetic sequence)

$\;\;\underline{\textbf{\textsf{ To Find :-}}}$

• The middle term of the sequence

• The angles of the triangle

$\;\;\underline{\textbf{\textsf{ Solution :-}}}$

Let the angles of the triangle be a - d , a , a + d

Then , from the properties of triangle will be :-

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

$\underline{\textsf{ Therefore, middle term of the AP is \textbf{ 60°}}}.$

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Again, it is given that the triangle is a right angled triangle. It seems that the largest angle is 90°.

Then,

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

$\underline{\textsf{ Hence, common difference of the AP is \textbf{ 30°}}}.$

The other angle of the triangle will be :-

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

$\;\;\underline{\textbf{\textsf{ Hence-}}}$

$\underline{\textsf{ The angles of the triangle are \textbf{ 90°, 60° and 30°}}}.$

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Answer 2

Mate your answer : 90°, 60° and 30°