Question

3.
triangle pqr with right angle at q, angles at p and r are 45° each, hypotenuse pr = 3√2. find pq and qr.
4.
triangle hgi with right angle at i, angles at h and g are 45° each, leg hi = 5. find gi and hg.
5.
triangle abc with right angle at a, angles at b and c are 45° each, leg bc = 20. find ab and ac.

Explanation:

Problem 3:

Step1: Identify triangle type

This is a 45-45-90 right triangle, so legs are equal, hypotenuse $= \text{leg} \times \sqrt{2}$.

Step2: Calculate leg length

Let $PQ = QR = x$. Hypotenuse $PR = 3\sqrt{2}$, so $x\sqrt{2} = 3\sqrt{2}$. Solve for $x$:
$x = \frac{3\sqrt{2}}{\sqrt{2}} = 3$

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Problem 4:

Step1: Identify triangle type

This is a 45-45-90 right triangle, legs are equal, hypotenuse $= \text{leg} \times \sqrt{2}$.

Step2: Find equal leg

$HI = GI = 5$ (legs of 45-45-90 triangle are equal).

Step3: Calculate hypotenuse $HG$

$HG = 5 \times \sqrt{2} = 5\sqrt{2}$

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Problem 5:

Step1: Identify triangle type

This is a 45-45-90 right triangle, legs are equal, hypotenuse $= \text{leg} \times \sqrt{2}$.

Step2: Calculate leg length

Let $AB = AC = x$. Hypotenuse $BC = 20$, so $x\sqrt{2} = 20$. Solve for $x$:
$x = \frac{20}{\sqrt{2}} = 10\sqrt{2}$

Answer:

1. $PQ = 3$, $QR = 3$
2. $GI = 5$, $HG = 5\sqrt{2}$
3. $AB = 10\sqrt{2}$, $AC = 10\sqrt{2}$