To find the perimeter of the triangle, we need to add up the lengths of all three sides.

The triangle has:

1. A vertical height of 5 cm,
2. A base split into 3 cm and 12 cm (total = 15 cm),
3. Two unknown slant sides (hypotenuses) that need to be calculated using Pythagoras' theorem.

Step 1: Split the Triangle

The triangle is right-angled, so we can split it as:

• Left triangle (base 3 cm, height 5 cm)
• Right triangle (base 12 cm, height 5 cm)

Step 2: Calculate the Hypotenuses

For the left triangle: $\text{Hypotenuse} = \sqrt{(3)^2 + (5)^2} = \sqrt{9 + 25} = \sqrt{34} \approx 5.8 \, \text{cm}$

For the right triangle: $\text{Hypotenuse} = \sqrt{(12)^2 + (5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \, \text{cm}$

Step 3: Find the Perimeter

Now, add all three sides of the triangle:

• Left hypotenuse ≈ 5.8 cm
• Right hypotenuse = 13 cm
• Base = 15 cm

$\text{Perimeter} = 5.8 + 13 + 15 = 33.8 \, \text{cm}$

Final Answer:

The perimeter of the triangle is 33.8 cm (to 1 decimal place).

Do you want me to explain any step in more detail? Here are 5 related questions to explore further:

1. How do you calculate the hypotenuse using Pythagoras' theorem?
2. How is the perimeter of a right-angled triangle determined?
3. What is the area of this triangle?
4. Can Pythagoras' theorem be applied to non-right-angled triangles?
5. How would the solution change if the triangle were drawn differently?

Tip: Always double-check the units and rounding when solving perimeter or area problems.