Understanding Transformations in Year 10 Math

In Year 10 Math, learning about transformations helps us show that figures are congruent.

Congruent means that shapes are the same in size and form, even if they are facing different ways. Let’s look at how transformations help us prove this.

Different Types of Transformations

1. Translation: This is when we slide a shape without changing its size or shape.

   • For example, if we move triangle ABC to triangle A'B'C', they will still be congruent because we didn’t change the triangle.

2. Rotation: This means turning a shape around a fixed point.

   • If we rotate triangle DEF by 90 degrees around point O to match triangle D'E'F', they remain congruent.

3. Reflection: This is flipping a shape over a line.

   • For example, if we flip square GHIJ over line XY to create square G'H'I'J', they are still congruent even though one looks like a mirror image of the other.

Proving Congruence with Transformations

To show that two shapes are congruent, follow these simple steps:

1. Identify the Figures: Start with the shapes you want to compare, like triangle PQR and triangle P'Q'R'.

2. Apply Transformations: Use one of the transformations (translation, rotation, or reflection) to see if you can fit one figure over the other.

3. Match Vertices and Sides: If all the matching points (corners) of both shapes line up perfectly after your transformations, you have shown that the shapes are congruent.

Example

Let’s look at two triangles:

• Triangle PQR has points at (0,0), (2,0), and (1,√3).
• Triangle P'Q'R' has points at (1,√3), (3,√3), and (2,0).

If we translate triangle PQR to the right by 1 unit, it lines up perfectly with triangle P'Q'R'. This means triangle PQR is congruent to triangle P'Q'R'.

Conclusion

By moving and changing figures using transformations, we can easily show that they are congruent. This helps us understand important ideas in geometry!