https://r2.erweima.ai/mathbot/image/1725523338653yq0a3odj.jpg

Let's work through the tasks presented in the image.

### Task 1: Missing Side Lengths
Here, you need to calculate the missing side lengths for the triangles using the Pythagorean theorem. For a right-angled triangle, the formula is:
\[
c^2 = a^2 + b^2
\]
where \(c\) is the hypotenuse, and \(a\) and \(b\) are the other two sides.

- **(a)** \(a = 9 \, \text{cm}, b = 12 \, \text{cm}\) → \(c = ?\)
  \[
  c^2 = 9^2 + 12^2 = 81 + 144 = 225 \quad \Rightarrow \quad c = \sqrt{225} = 15 \, \text{cm}
  \]
- **(b)** \(b = 12 \, \text{cm}, c = 13 \, \text{cm}\) → \(a = ?\)
  \[
  a^2 + 12^2 = 13^2 \quad \Rightarrow \quad a^2 + 144 = 169 \quad \Rightarrow \quad a^2 = 25 \quad \Rightarrow \quad a = \sqrt{25} = 5 \, \text{cm}
  \]
- **(c)** \(a = 7 \, \text{cm}, c = 25 \, \text{cm}\) → \(b = ?\)
  \[
  7^2 + b^2 = 25^2 \quad \Rightarrow \quad 49 + b^2 = 625 \quad \Rightarrow \quad b^2 = 576 \quad \Rightarrow \quad b = \sqrt{576} = 24 \, \text{cm}
  \]
- **(d)** \(a = 22 \, \text{cm}, c = 53 \, \text{cm}\) → \(b = ?\)
  \[
  22^2 + b^2 = 53^2 \quad \Rightarrow \quad 484 + b^2 = 2809 \quad \Rightarrow \quad b^2 = 2325 \quad \Rightarrow \quad b = \sqrt{2325} \approx 48.22 \, \text{cm}
  \]

### Task 2: Triangle Sides
Here you have to complete a table by finding missing values for the sides \(a\), \(b\), and \(c\) of several triangles. The data in the table seems incomplete, so you would apply the Pythagorean theorem or use other geometric properties to find the missing values. 

#### Right-angled triangles:
For example, if two sides of a right triangle are given, you can calculate the third one similarly to Task 1.

### Task 3: Distances in a Triangle
The problem asks to find the distances between three points \(A(1, 2)\), \(B(3, -1)\), and \(C(-5, 2)\).

- The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
  \[
  d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
  \]
  
  Let's calculate the distances:

  - \(d_{AB}\):
  \[
  d = \sqrt{(3 - 1)^2 + (-1 - 2)^2} = \sqrt{(2)^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.61 \, \text{units}
  \]

  - \(d_{BC}\):
  \[
  d = \sqrt{(-5 - 3)^2 + (2 - (-1))^2} = \sqrt{(-8)^2 + (3)^2} = \sqrt{64 + 9} = \sqrt{73} \approx 8.54 \, \text{units}
  \]

  - \(d_{CA}\):
  \[
  d = \sqrt{(-5 - 1)^2 + (2 - 2)^2} = \sqrt{(-6)^2 + (0)^2} = \sqrt{36} = 6 \, \text{units}
  \]

  To check if the triangle is right-angled, apply the Pythagorean theorem to the distances. 

### Task 4: Trapezoid Side Lengths
Here, you need to calculate the unknown side lengths in trapezoids using given information. Applying basic geometric properties and possibly the Pythagorean theorem will help solve these.

### Task 5: Lengths in a Triangle
In this task, you need to determine two unknown lengths \(x\) and \(y\) inside triangles.

Would you like detailed steps for specific tasks, or further clarification on a particular problem?

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**Follow-up questions:**
1. How do you calculate distances between two points in a coordinate plane?
2. What is the significance of the Pythagorean theorem in geometry?
3. Can you identify the properties of a right-angled triangle?
4. How would you approach solving for side lengths in non-right-angled triangles?
5. What are some methods to determine if a triangle is right-angled based on its side lengths?

**Tip:** Always try to sketch a diagram when working with geometric problems. It can make visualizing the relationships between sides and angles much clearer!

Calculate missing side lengths, distances between points, and unknown trapezoid and triangle dimensions from geometric problems.

Learn how to calculate missing side lengths in triangles, distances between points, and unknown trapezoid side lengths using the Pythagorean theorem and distance formula. Step-by-step solutions for students in Grades 6-8.

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