Let \(m, n\) and \(o\) be the side lengths of a triangle, with \(o\) the longest side.
i) State the Baudhayana -Pythagoras theorem and its converse in "if-then" form.
ii) Prove the converse by constructing a right triangle with perpendicular sides \(m\) and \(n\) using the SSS congruence criterion.
iii) Use the converse to determine whether a triangle with side lengths \(8\ cm, 15cm\) and \(17\ cm\) is right angled triangle.
i) Theorem: If a triangle is right-angled with \(o\) the longest side then
Converse : If sides of the triangle are \(m, n\) and \(o\) which satisfies , then the triangle is a right-angled triangle.
ii) In \(\triangle ABC\), if \(BC\ =\ m\), \(AC = n\) and \(AB\ =\ o\) and
Construct \(\triangle XYZ\) with \(YZ\ =\ m\), \(XZ\ =\ n\) and \(\angle Z = 90 ^\circ \).
By pythagoras theorem in \(\triangle XYZ\),
So, \(XY\ =\)
Thus, \(BC\ =\) , \(AC\ =\) and \(AB\ =\).
Therefore \(\triangle ABC \cong \triangle \) by SSS congurence criterion.
By CPCT \(\angle C\ =\ \angle \) \( =\) \(^\circ \).
Hence proved.
iii) Side length are \(8\ cm, 15cm\) and \(17\ cm\)
Therefore, by converse of pythagoras theorem it is right-angled triangle.