To construct an isosceles triangle with a base of 8 cm and a height of 4 cm, follow these steps:

Step 1: Construct the Isosceles Triangle

1. Draw the Base: Draw a horizontal line segment 8 cm long. Label the endpoints as $A$ and $B$.
2. Find the Midpoint: Mark the midpoint $M$ of segment $AB$ (which is 4 cm from both $A$ and $B$).
3. Draw the Height: From point $M$, draw a vertical line segment $MC$ of length 4 cm. Label the top point as $C$.
4. Connect the Points: Draw line segments $AC$ and $BC$ to complete the triangle $ABC$.

Step 2: Construct the Similar Triangle

To create a triangle similar to triangle $ABC$ with sides that are $\frac{2}{3}$ the length of the original triangle:

1. Determine the Side Lengths:

   • The length of $AC$ and $BC$ can be calculated using the Pythagorean theorem: $AC = BC = \sqrt{(4^2 + 4^2)} = \sqrt{32} \approx 5.66 \text{ cm}$
   • So, the sides of triangle $ABC$ are approximately $8 \text{ cm}$ (base) and $5.66 \text{ cm}$ (sides).

2. Calculate the Similar Triangle Dimensions:

   • Base: $\frac{2}{3} \times 8 \text{ cm} = \frac{16}{3} \text{ cm} \approx 5.33 \text{ cm}$
   • Height: $\frac{2}{3} \times 4 \text{ cm} = \frac{8}{3} \text{ cm} \approx 2.67 \text{ cm}$
   • Side Lengths: $\frac{2}{3} \times 5.66 \text{ cm} \approx 3.77 \text{ cm}$

Step 3: Construct the Similar Triangle

1. Draw the New Base: Draw a horizontal line segment approximately $5.33 \text{ cm}$ long. Label the endpoints as $D$ and $E$.
2. Find the Midpoint: Mark the midpoint $N$ of segment $DE$.
3. Draw the Height: From point $N$, draw a vertical line segment of length $\frac{8}{3} \text{ cm}$ (approximately 2.67 cm). Label the top point as $F$.
4. Connect the Points: Draw line segments $DF$ and $EF$ to complete the smaller triangle $DEF$.

Summary

You now have two isosceles triangles: the original $ABC$ and the smaller, similar triangle $DEF$.

If you need further details or clarifications, let me know! Here are some related questions for you to consider:

1. What is the relationship between the angles of similar triangles?
2. How can you prove that two triangles are similar?
3. What are the properties of isosceles triangles?
4. Can you describe how to calculate the area of these triangles?
5. How does changing the height affect the triangle's properties?

Tip: When constructing triangles, always use precise measurements for accuracy.