The triangle (a) is the right angled triangle. Option (a) is correct.
Further explanation:
The Pythagorean formula can be expressed as,
![\boxed{{H^2} = {P^2} + {B^2}}.](https://tex.z-dn.net/?f=%5Cboxed%7B%7BH%5E2%7D%20%3D%20%7BP%5E2%7D%20%2B%20%7BB%5E2%7D%7D.)
Here, H represents the hypotenuse, P represents the perpendicular and B represents the base.
Given:
Explanation:
In triangle (a),
The hypotenuse is ![\sqrt {13}](https://tex.z-dn.net/?f=%5Csqrt%20%7B13%7D)
The base is 3.
The perpendicular is 2.
Use the Pythagorean Theorem in triangle (a),
![\begin{aligned}{2^2} + {3^2} &= {\left( {\sqrt {13} } \right)^2}\\4 + 9 &= 13\\13 &=13\\\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%7B2%5E2%7D%20%2B%20%7B3%5E2%7D%20%26%3D%20%7B%5Cleft%28%20%7B%5Csqrt%20%7B13%7D%20%7D%20%5Cright%29%5E2%7D%5C%5C4%20%2B%209%20%26%3D%2013%5C%5C13%20%26%3D13%5C%5C%5Cend%7Baligned%7D)
Triangle (a) is right angled.
In triangle (b),
The hypotenuse is 5.
The base is ![3\sqrt 2.](https://tex.z-dn.net/?f=3%5Csqrt%202.)
The perpendicular is 2.
Use the Pythagorean Theorem in triangle (b),
![\begin{aligned}{2^2} + {\left( {3\sqrt 2 } \right)^2} &= {5^2}\\4 + 18&= 25\\ 22 &\ne 25\\\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%7B2%5E2%7D%20%2B%20%7B%5Cleft%28%20%7B3%5Csqrt%202%20%7D%20%5Cright%29%5E2%7D%20%26%3D%20%7B5%5E2%7D%5C%5C4%20%2B%2018%26%3D%2025%5C%5C%2022%20%26%5Cne%2025%5C%5C%5Cend%7Baligned%7D)
Triangle (b) is not right angled.
In triangle (c),
The hypotenuse is ![\sqrt {43}.](https://tex.z-dn.net/?f=%5Csqrt%20%7B43%7D.)
The base is ![3\sqrt 3.](https://tex.z-dn.net/?f=3%5Csqrt%203.)
The perpendicular is 2.
Use the Pythagorean Theorem in triangle (b),
![\begin{aligned}{2^2} + {\left( {3\sqrt 3 } \right)^2}&= {\left( {\sqrt {43} } \right)^2}\\4 + 27 &= 43\\31 &\ne 43\\\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%7B2%5E2%7D%20%2B%20%7B%5Cleft%28%20%7B3%5Csqrt%203%20%7D%20%5Cright%29%5E2%7D%26%3D%20%7B%5Cleft%28%20%7B%5Csqrt%20%7B43%7D%20%7D%20%5Cright%29%5E2%7D%5C%5C4%20%2B%2027%20%26%3D%2043%5C%5C31%20%26%5Cne%2043%5C%5C%5Cend%7Baligned%7D)
Triangle (c) is not right angled.
The triangle (a) is the right angled triangle. Option (a) is correct.
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Answer details:
Grade: High School
Subject: Mathematics
Chapter: Triangles
Keywords: triangles, right angled triangle, Pythagoras Theorem, perpendicular, base, hypotenuse.