Let's go through each of the questions in the image and solve them step by step.

1. Find the length of side AB in each figure:

a. Right triangle with sides:

• $CD = 10 \, \text{cm}$
• $DA = 6 \, \text{cm}$
• $BC = 15 \, \text{cm}$

We can use the Pythagorean theorem: $AB^2 = BC^2 + AC^2$ First, find $AC$ using Pythagoras on $\triangle ACD$: $AC^2 = CD^2 - DA^2 = 10^2 - 6^2 = 100 - 36 = 64 \quad \Rightarrow \quad AC = 8 \, \text{cm}$ Now, use the Pythagorean theorem on $\triangle ABC$: $AB^2 = BC^2 + AC^2 = 15^2 + 8^2 = 225 + 64 = 289 \quad \Rightarrow \quad AB = \sqrt{289} = 17 \, \text{cm}$

b. Right triangle with sides:

• $AE = 12 \, \text{cm}$
• $AC = 15 \, \text{cm}$
• $BC = 20 \, \text{cm}$

Using the Pythagorean theorem: $AB^2 = AC^2 + BC^2 = 15^2 + 20^2 = 225 + 400 = 625 \quad \Rightarrow \quad AB = \sqrt{625} = 25 \, \text{cm}$

2. Find the area of each figure:

a. Trapezoid with:

• Bases: $13 \, \text{cm}$ and $25 \, \text{cm}$
• Height: $24 \, \text{cm}$

The formula for the area of a trapezoid is: $\text{Area} = \frac{1}{2} \times ( \text{Base}_1 + \text{Base}_2 ) \times \text{Height}$ Substituting the values: $\text{Area} = \frac{1}{2} \times (13 + 25) \times 24 = \frac{1}{2} \times 38 \times 24 = 456 \, \text{cm}^2$

b. Rhombus with:

• Diagonals: $12 \, \text{cm}$ and $12 \, \text{cm}$

The formula for the area of a rhombus is: $\text{Area} = \frac{1}{2} \times d_1 \times d_2$ Substituting the values: $\text{Area} = \frac{1}{2} \times 12 \times 12 = 72 \, \text{cm}^2$

3. Determine which set of numbers forms the sides of a right triangle:

We use the Pythagorean theorem to check which group of numbers satisfies $a^2 + b^2 = c^2$.

• Set a: $12, 16, 20$ $12^2 + 16^2 = 144 + 256 = 400 \quad \text{and} \quad 20^2 = 400 \quad \Rightarrow \quad \text{This set forms a right triangle.}$

• Set b: $5, 6, 7$ $5^2 + 6^2 = 25 + 36 = 61 \quad \text{and} \quad 7^2 = 49 \quad \Rightarrow \quad \text{This set does not form a right triangle.}$

• Set c: $7, 12, 13$ $7^2 + 12^2 = 49 + 144 = 193 \quad \text{and} \quad 13^2 = 169 \quad \Rightarrow \quad \text{This set does not form a right triangle.}$

So, the correct answer is Set a: 12, 16, 20.

Would you like more detailed steps or explanations on any of these problems?

Here are 5 related questions to extend your understanding:

1. How do you calculate the hypotenuse of a right triangle?
2. What is the formula for the area of a trapezoid?
3. How can you check if three sides form a right triangle using the Pythagorean theorem?
4. What is the relationship between diagonals and area in a rhombus?
5. Can the Pythagorean theorem be used for non-right triangles?

Tip: When solving geometry problems, it's helpful to sketch diagrams and label all given dimensions clearly.