Question

in △pqr, what is the length of segment qr? triangle diagram: right angle at q, ∠p=45°, ∠r=45°, hypotenuse pr=22 options: (22sqrt{3}), (11), (11sqrt{2}), (22sqrt{2})

Explanation:

Step1: Identify triangle type

$\triangle PQR$ is a right - isosceles triangle (since $\angle Q = 90^{\circ}$, $\angle P=\angle R = 45^{\circ}$), so $PQ = QR$. Let $PQ = QR=x$.

Step2: Apply Pythagorean theorem

In right - triangle $PQR$, by Pythagorean theorem, $PQ^{2}+QR^{2}=PR^{2}$. Since $PQ = QR=x$ and $PR = 22$, we have $x^{2}+x^{2}=22^{2}$.
Simplify the left - hand side: $2x^{2}=484$.
Divide both sides by 2: $x^{2}=242$.
Take the square root of both sides: $x=\sqrt{242}=\sqrt{121\times2}=11\sqrt{2}$? Wait, no, wait. Wait, if we use the property of 45 - 45 - 90 triangle, the hypotenuse $c$ and the legs $a$ (since $a = b$ in 45 - 45 - 90 triangle) are related by $c=a\sqrt{2}$, so $a=\frac{c}{\sqrt{2}}$.
Given $c = 22$, then $a=\frac{22}{\sqrt{2}}=\frac{22\sqrt{2}}{2}=11\sqrt{2}$? Wait, no, wait, I made a mistake. Wait, in a 45 - 45 - 90 triangle, the legs are equal, and hypotenuse $=leg\times\sqrt{2}$. So if hypotenuse $PR = 22$, then $leg= \frac{hypotenuse}{\sqrt{2}}=\frac{22}{\sqrt{2}}=\frac{22\sqrt{2}}{2}=11\sqrt{2}$? Wait, no, wait, let's re - do.
Wait, the triangle has $\angle Q = 90^{\circ}$, $\angle P = 45^{\circ}$, $\angle R=45^{\circ}$, so it's an isosceles right triangle with $PQ = QR$. Let $PQ = QR = x$. Then by Pythagoras: $x^{2}+x^{2}=22^{2}\implies2x^{2}=484\implies x^{2}=242\implies x=\sqrt{242}=\sqrt{121\times2}=11\sqrt{2}$? Wait, no, $\sqrt{242}=\sqrt{121\times2}=11\sqrt{2}$? Wait, 121*2 = 242, yes. But wait, maybe I misread the options. Wait, the options are $22\sqrt{3}$, 11, $11\sqrt{2}$, $22\sqrt{2}$. Wait, no, wait, maybe the hypotenuse is not 22? Wait, no, the side $PR$ is 22. Wait, no, in a 45 - 45 - 90 triangle, the legs are equal, and hypotenuse $=leg\sqrt{2}$. So if we let the leg be $x$, then hypotenuse $x\sqrt{2}=22\implies x=\frac{22}{\sqrt{2}} = 11\sqrt{2}$. Wait, but let's check again.
Wait, maybe I made a mistake in the triangle's angle. Wait, $\angle Q$ is right angle, $\angle P = 45^{\circ}$, $\angle R = 45^{\circ}$, so it's an isosceles right triangle. So legs $PQ$ and $QR$ are equal, hypotenuse $PR = 22$. So by the formula for 45 - 45 - 90 triangle: leg $=\frac{hypotenuse}{\sqrt{2}}=\frac{22}{\sqrt{2}}=\frac{22\sqrt{2}}{2}=11\sqrt{2}$. Wait, but the options have $11\sqrt{2}$ as one of them. Wait, but let's check the calculation again.
Wait, $x^{2}+x^{2}=22^{2}\implies2x^{2}=484\implies x^{2}=242\implies x=\sqrt{242}=\sqrt{121\times2}=11\sqrt{2}$. Yes, that's correct.

Answer:

$11\sqrt{2}$