To construct a triangle ABC where side $AB = 3 \, \text{cm}$, side $AC = 5 \, \text{cm}$, and $\angle BAC = 90^\circ$, follow these steps:

Steps for Construction:

1. Draw Line Segment AB:
   Draw a horizontal line segment $AB = 3 \, \text{cm}$.

2. Construct a $90^\circ$ Angle at Point A:
   At point $A$, use a protractor to construct a $90^\circ$ angle. This means the line will be perpendicular to line $AB$.

3. Mark Point C on the Perpendicular Line:
   From point $A$, measure a distance of $AC = 5 \, \text{cm}$ along the perpendicular line. Mark this point as $C$.

4. Connect Point C to Point B:
   Draw a line connecting $C$ to $B$. Now you have your required triangle $\triangle ABC$.

Verifying the Triangle:

This triangle is a right triangle with $\angle BAC = 90^\circ$. By the Pythagorean theorem, check if: $AB^2 + AC^2 = BC^2$ $3^2 + 5^2 = BC^2 \quad \text{or} \quad 9 + 25 = 34$ Thus, $BC = \sqrt{34} \approx 5.83 \, \text{cm}$.

Would you like more details, or do you have any questions?

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Here are 5 related questions to explore:

1. How do you verify the angle in triangle constructions?
2. Can you construct a triangle given three sides (SSS criterion)?
3. How do you apply the Pythagorean theorem to other right triangles?
4. What is the general formula for calculating the area of a right triangle?
5. How would you construct a triangle given one angle and two sides not including that angle?

Tip: Always ensure the accuracy of your angle constructions using a protractor to maintain geometric precision.