# Which is Acute,Obtuse, and Right?(3, 5,and 7)(4.5, 6, and 7.5)(6, 8, and 9)(1.2, 2.5, and 3.1)(8, 15, and 17)(14, 19, and 21)

Mar 17, 2017

#### Explanation:

Although it has not been clearly mentioned, it appears that questioner has given the lengths of sides of $\textcolor{red}{\text{six}}$ triangles.

Let these be

$\Delta A - \left(3 , 5 , 7\right)$
$\Delta B - \left(4.5 , 6 , 7.5\right)$
$\Delta C - \left(6 , 8 , 9\right)$
$\Delta D - \left(1.2 , 2.5 , 3.1\right)$
$\Delta E - \left(8 , 15 , 17\right)$
$\Delta F - \left(14 , 19 , 21\right)$

In a right angled triangle say $\Delta - \left(P , Q , R\right)$, if $R$ is the largest side, then ${R}^{2} = {P}^{2} + {Q}^{2}$.

In an acute angled triangle say $\Delta - \left(P , Q , R\right)$, if $R$ is the largest side, then ${R}^{2} < {P}^{2} + {Q}^{2}$.

In an obtuse angled triangle say $\Delta - \left(P , Q , R\right)$, if $R$ is the largest side, then ${R}^{2} > {P}^{2} + {Q}^{2}$.

Hence as ${7}^{2} > {3}^{2} + {5}^{2}$, $\Delta A$ is obtuse angled triangle.

As ${7.5}^{2} = {4.5}^{2} + {6}^{2}$, $\Delta B$ is right angled triangle.

As ${9}^{2} < {6}^{2} + {8}^{2}$, $\Delta C$ is acute angled triangle.

As ${3.1}^{2} > {1.2}^{2} + {2.5}^{2}$, $\Delta D$ is obtuse angled triangle.

As ${17}^{2} = {8}^{2} + {15}^{2}$, $\Delta E$ is right angled triangle.

As ${21}^{2} > {9}^{2} + {14}^{2}$, $\Delta F$ is obtuse angled triangle.