Let's solve each part of the problem based on the provided diagrams.

a. Determine the length of side $AB$

In the first diagram (a), we have a trapezoid, and we need to find side $AB$. The given information is:

• $CD = 10 \, \text{cm}$ (height)
• $AD = 6 \, \text{cm}$ (one side of the right triangle at $A$)
• $BC = 15 \, \text{cm}$ (upper base of the trapezoid)

Looking at the right triangle $\triangle ABC$, where $AC$ is the hypotenuse:

• $AB$ can be found using the Pythagorean theorem.

$AB^2 = BC^2 + CD^2$

Substituting the values:

$AB^2 = 15^2 + 10^2$ $AB^2 = 225 + 100 = 325$ $AB = \sqrt{325} \approx 18.03 \, \text{cm}$

So, the length of side $AB$ is approximately $18.03 \, \text{cm}$.

b. Determine the length of side $AB$

In the second diagram (b), we have a triangle, and we need to find side $AB$. The given information is:

• $AC = 15 \, \text{cm}$ (one side of the triangle)
• $BC = 20 \, \text{cm}$ (another side of the triangle)
• $CE = 12 \, \text{cm}$ (height of the triangle from vertex $C$)

To find $AB$, we use the fact that it's a right triangle $\triangle AEB$. Again, applying the Pythagorean theorem:

$AB^2 = AC^2 + BC^2$

Substituting the values:

$AB^2 = 15^2 + 20^2 = 225 + 400 = 625$ $AB = \sqrt{625} = 25 \, \text{cm}$

So, the length of side $AB$ is $25 \, \text{cm}$.

Would you like further clarification on any of these steps or more details?

Related questions:

1. How is the Pythagorean theorem used to calculate the sides of a triangle?
2. Can the Pythagorean theorem be used in non-right triangles?
3. What are the properties of trapezoids that help in solving geometric problems?
4. How do you calculate the area of a trapezoid or a triangle?
5. What other methods can be used to solve for unknown sides in geometric shapes?

Tip: When solving problems with right triangles, always check if you can apply the Pythagorean theorem for efficient calculation!