Let's analyze the image and solve the given problem step by step.

Problem:

We need to find the areas of the following from the figure:

1. $\triangle ABC$
2. $\triangle ACD$
3. Quadrilateral $ABCD$

Given:

• $AB = 6$, $BC = 8$, and $AC$ (hypotenuse) can be calculated.
• $AD = 10$, $CD = 10$.

Step 1: Calculate $\triangle ABC$

In $\triangle ABC$, $AB$ and $BC$ form a right angle. The formula for the area of a triangle is: $\text{Area of a triangle} = \frac{1}{2} \times \text{base} \times \text{height}$ Here, base = $BC = 8$, and height = $AB = 6$. Therefore: $\text{Area of } \triangle ABC = \frac{1}{2} \times 8 \times 6 = 24 \, \text{square units}.$

Step 2: Calculate $\triangle ACD$

In $\triangle ACD$, the sides are given as $AD = 10$, $CD = 10$, and $AC$ acts as the base of this isosceles triangle. We need $AC$ first.

Using Pythagoras theorem in $\triangle ABC$:

$AC = \sqrt{AB^2 + BC^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10.$ So, $AC = 10$.

Find the height of $\triangle ACD$:

In an isosceles triangle, the altitude divides the base into two equal parts. Hence, $AC = 10$ is divided into two parts of $5$ each. Using the Pythagoras theorem for one of the smaller right triangles formed: $\text{Height of } \triangle ACD = \sqrt{AD^2 - \left(\frac{AC}{2}\right)^2} = \sqrt{10^2 - 5^2} = \sqrt{100 - 25} = \sqrt{75} = 5\sqrt{3}.$

Calculate the area:

$\text{Area of } \triangle ACD = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10 \times 5\sqrt{3} = 25\sqrt{3} \, \text{square units}.$

Step 3: Calculate Quadrilateral $ABCD$

The area of $ABCD$ is the sum of the areas of $\triangle ABC$ and $\triangle ACD$: $\text{Area of } ABCD = \text{Area of } \triangle ABC + \text{Area of } \triangle ACD = 24 + 25\sqrt{3} \, \text{square units}.$

Final Answers:

1. $\text{Area of } \triangle ABC = 24 \, \text{square units}$
2. $\text{Area of } \triangle ACD = 25\sqrt{3} \, \text{square units}$
3. $\text{Area of quadrilateral } ABCD = 24 + 25\sqrt{3} \, \text{square units}$

Let me know if you need detailed steps or clarification on any part!

Follow-up Questions:

1. How do you derive the height of an isosceles triangle?
2. What is the significance of the Pythagorean theorem in this problem?
3. Can you calculate the perimeter of $ABCD$?
4. What are the properties of isosceles triangles used in this problem?
5. How can the area be verified using another method?

Tip:

Always check if the given figure involves symmetry or special properties (like right angles or isosceles triangles) to simplify calculations!