Question

find the unknown side lengths in each right triangle. 1. ab = bc = 2. qr = pq = 3. lk = jl = 4. a = b = 5. v = u =

Explanation:

All these are isosceles right triangles (45-45-90 triangles), where the legs are equal, and the hypotenuse is $\text{leg} \times \sqrt{2}$.

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Triangle 1:

Step1: Identify hypotenuse, find legs

Given hypotenuse $AC=18$. For 45-45-90 triangles, $\text{leg} = \frac{\text{hypotenuse}}{\sqrt{2}}$
$\text{AB} = \text{BC} = \frac{18}{\sqrt{2}} = 9\sqrt{2}$

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Triangle 2:

Step1: Find equal leg QR

Leg $PN=5\sqrt{16}=20$. In 45-45-90 triangle, legs are equal:
$QR = 20$

Step2: Calculate hypotenuse PQ

$\text{Hypotenuse} = \text{leg} \times \sqrt{2}$
$PQ = 20\sqrt{2}$

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Triangle 3:

Step1: Find equal leg LK

Leg $JK=4\sqrt{3}$. In 45-45-90 triangle, legs are equal:
$LK = 4\sqrt{3}$

Step2: Calculate hypotenuse JL

$\text{Hypotenuse} = \text{leg} \times \sqrt{2}$
$JL = 4\sqrt{3} \times \sqrt{2} = 4\sqrt{6}$

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Triangle 4:

Step1: Find leg b from hypotenuse

Given hypotenuse $2\sqrt{2}$. $\text{leg} = \frac{\text{hypotenuse}}{\sqrt{2}}$
$b = \frac{2\sqrt{2}}{\sqrt{2}} = 2$

Step2: Identify equal leg a

In 45-45-90 triangle, legs are equal:
$a = 2$

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Triangle 5:

Step1: Find equal leg u

Leg $v=\frac{3\sqrt{2}}{4}$. In 45-45-90 triangle, legs are equal:
$u = \frac{3\sqrt{2}}{4}$

Step2: Calculate hypotenuse (already given as v's pair, confirm leg equality)

$v = \frac{3\sqrt{2}}{4}$

Answer:

1. $AB = 9\sqrt{2}$, $BC = 9\sqrt{2}$
2. $QR = 20$, $PQ = 20\sqrt{2}$
3. $LK = 4\sqrt{3}$, $JL = 4\sqrt{6}$
4. $a = 2$, $b = 2$
5. $v = \frac{3\sqrt{2}}{4}$, $u = \frac{3\sqrt{2}}{4}$