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: $PQ = 7, QR = 24, PR = 25$.
(D) In $\Delta PQR$,the side lengths are $PQ = 7$,$QR = 24$,and $PR = 25$.
The longest side is $PR = 25$.
Calculate the square of the longest side:
$PR^2 = 25^2 = 625$
Calculate the sum of the squares of the other two sides:
$PQ^2 + QR^2 = 7^2 + 24^2 = 49 + 576 = 625$
Since $PQ^2 + QR^2 = PR^2$,the triangle satisfies the converse of the Pythagoras theorem.
Therefore,$\Delta PQR$ is a right-angled triangle,and the angle opposite to the hypotenuse $PR$,which is $\angle Q$,is the right angle.
The longest side is $PR = 25$.
Calculate the square of the longest side:
$PR^2 = 25^2 = 625$
Calculate the sum of the squares of the other two sides:
$PQ^2 + QR^2 = 7^2 + 24^2 = 49 + 576 = 625$
Since $PQ^2 + QR^2 = PR^2$,the triangle satisfies the converse of the Pythagoras theorem.
Therefore,$\Delta PQR$ is a right-angled triangle,and the angle opposite to the hypotenuse $PR$,which is $\angle Q$,is the right angle.
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View SolutionWhich of the following correctly matches the information in Part $I$ and Part $II$?
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| Part $I$ | Part $II$ |
| $1.$ In $\Delta ABC$ and $\Delta PQR, \angle A \cong \angle P$ and $\angle C \cong \angle Q$ | $a.$ Correspondence $ABC \leftrightarrow RQP$ is a similarity. |
| $2.$ In $\Delta ABC$ and $\Delta PQR, \frac{AB}{QR} = \frac{BC}{PQ}$ and $\angle B \cong \angle Q$ | $b.$ Correspondence $ABC \leftrightarrow QPR$ is a similarity. |
| $3.$ In $\Delta ABC$ and $\Delta PQR, \frac{AB}{PQ} = \frac{BC}{PR} = \frac{CA}{QR}$ | $c.$ Correspondence $ABC \leftrightarrow PQR$ is a similarity. |
| $4.$ In $\Delta ABC$ and $\Delta PQR, \frac{AB}{PQ} = \frac{CA}{PR}$ and $\angle A \cong \angle P$ | $d.$ Correspondence $ABC \leftrightarrow PRQ$ is a similarity. |
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