Below are given the measures of sides $\overline{PQ}$,$\overline{QR}$ and $\overline{PR}$ of $\Delta PQR$. In each case,determine whether $\Delta PQR$ is a right-angled triangle or not. If it is a right-angled triangle,state which angle is a right angle: $PQ = 8, QR = 6, PR = 12$.
(D) In $\Delta PQR$,the side lengths are $PQ = 8$,$QR = 6$,and $PR = 12$.
The longest side is $\overline{PR}$.
Calculate the square of the longest side: $PR^2 = 12^2 = 144$.
Calculate the sum of the squares of the other two sides: $PQ^2 + QR^2 = 8^2 + 6^2 = 64 + 36 = 100$.
According to the converse of the Pythagoras theorem,a triangle is right-angled if the square of the longest side is equal to the sum of the squares of the other two sides.
Here,$PQ^2 + QR^2 = 100$ and $PR^2 = 144$.
Since $PQ^2 + QR^2 \neq PR^2$,the triangle does not satisfy the condition for a right-angled triangle.
Therefore,$\Delta PQR$ is not a right-angled triangle.
The longest side is $\overline{PR}$.
Calculate the square of the longest side: $PR^2 = 12^2 = 144$.
Calculate the sum of the squares of the other two sides: $PQ^2 + QR^2 = 8^2 + 6^2 = 64 + 36 = 100$.
According to the converse of the Pythagoras theorem,a triangle is right-angled if the square of the longest side is equal to the sum of the squares of the other two sides.
Here,$PQ^2 + QR^2 = 100$ and $PR^2 = 144$.
Since $PQ^2 + QR^2 \neq PR^2$,the triangle does not satisfy the condition for a right-angled triangle.
Therefore,$\Delta PQR$ is not a right-angled triangle.
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