The image you've provided contains two triangles with the task to find the lengths of the missing sides for each triangle.

Triangle A:

Sides provided:

• $AB = 6$
• $BC = 8$
• $AC = ?$

Since this appears to be a right triangle (judging by the labeling and angles), we can use the Pythagorean theorem: $AB^2 + BC^2 = AC^2$ $6^2 + 8^2 = AC^2$ $36 + 64 = AC^2$ $100 = AC^2$ $AC = \sqrt{100} = 10$

So, the missing side $AC$ is $10$.

Triangle B:

Sides provided:

• $DE = 9$
• $EF = 12$
• $DF = ?$

Using the Pythagorean theorem again (assuming this is also a right triangle): $DE^2 + EF^2 = DF^2$ $9^2 + 12^2 = DF^2$ $81 + 144 = DF^2$ $225 = DF^2$ $DF = \sqrt{225} = 15$

So, the missing side $DF$ is $15$.

Summary of Results:

• For Triangle A: $AC = 10$
• For Triangle B: $DF = 15$

Would you like further details or explanations on any step?

Related Questions:

1. How would you solve for a missing side if the triangle wasn't a right triangle?
2. What are the properties of a right triangle?
3. How does the Pythagorean theorem apply to different types of triangles?
4. Can the Pythagorean theorem be used in three-dimensional space?
5. How can you verify if a triangle is a right triangle given side lengths?

Tip:

Always check if a triangle has a right angle before using the Pythagorean theorem; it only applies to right triangles!