G8a - Combinations of transformations - BossMaths.com - Free Printable

Let's solve this step-by-step based on the given exam-style question and the diagram.

We are given a right-angled triangle labeled P on a coordinate grid. Let's analyze each part carefully.

---

Step 1: Identify coordinates of shape P

From the graph:

Shape P is a right-angled triangle with vertices at:

- A = (2, 2)
- B = (5, 2)
- C = (5, 6)

So, it's a right triangle with the right angle at B = (5, 2), vertical side from (5,2) to (5,6), and horizontal base from (2,2) to (5,2).

---

## Part a) Rotate shape P by 180° about (1, 2). Label this rotated shape Q.

To rotate a point (x, y) by 180° about a center (a, b), the formula is:

> New point = (2a - x, 2b - y)

Here, center of rotation is (1, 2).

Apply this to each vertex of P:

Vertex A = (2, 2):

- x' = 2×1 - 2 = 2 - 2 = 0
- y' = 2×2 - 2 = 4 - 2 = 2
→ A' = (0, 2)

Vertex B = (5, 2):

- x' = 2×1 - 5 = 2 - 5 = -3
- y' = 2×2 - 2 = 4 - 2 = 2
→ B' = (-3, 2)

Vertex C = (5, 6):

- x' = 2×1 - 5 = -3
- y' = 2×2 - 6 = 4 - 6 = -2
→ C' = (-3, -2)

So, shape Q has vertices:
- (0, 2)
- (-3, 2)
- (-3, -2)

Plot these points: it’s a triangle pointing downward-left, same size as P.

---

## Part b) Reflect shape Q in the line x = 1. Label this reflected shape R.

Now reflect each point of Q over the vertical line x = 1.

Reflection over x = 1:
For any point (x, y), its reflection is:

> (2×1 - x, y) = (2 - x, y)

Apply to each vertex of Q:

A' = (0, 2):

- x' = 2 - 0 = 2 → (2, 2)

B' = (-3, 2):

- x' = 2 - (-3) = 5 → (5, 2)

C' = (-3, -2):

- x' = 2 - (-3) = 5 → (5, -2)

So, shape R has vertices:
- (2, 2)
- (5, 2)
- (5, -2)

Wait — that’s interesting.

Compare to original shape P:
- P: (2,2), (5,2), (5,6)
- R: (2,2), (5,2), (5,-2)

So R is like P but flipped vertically — same base, but instead of going up to (5,6), it goes down to (5,-2).

But we’ll use this for part c.

---

## Part c) Describe fully the single transformation that maps shape P onto R.

We need to find one transformation that takes P directly to R, without going through Q.

Let’s compare corresponding points:

Original P:
- A = (2, 2)
- B = (5, 2)
- C = (5, 6)

Final R:
- A → (2, 2) ← same!
- B → (5, 2) ← same!
- C → (5, -2)

So:
- (2,2) → (2,2)
- (5,2) → (5,2)
- (5,6) → (5,-2)

That means:
- The two bottom vertices stay fixed.
- The top vertex moves from (5,6) to (5,-2)

This is a reflection over the horizontal line y = 2, because:

- Midpoint between (5,6) and (5,-2) is:
- y = (6 + (-2))/2 = 4/2 = 2
- So the line of reflection is y = 2

And indeed:
- (5,6) reflected over y=2: distance is 4 units above → go 4 below → y = 2 - 4 = -2 → (5,-2) ✔
- (2,2): on the line → stays (2,2) ✔
- (5,2): on the line → stays (5,2) ✔

So yes! The transformation is a reflection in the line y = 2.

But wait — let’s double-check whether this matches the sequence.

We did:
- P → Q via 180° rotation about (1,2)
- Q → R via reflection in x=1

But now we’re being asked: what single transformation maps P → R?

We’ve just found that R is the reflection of P over y = 2.

Let’s verify:

- P: (2,2), (5,2), (5,6)
- Reflect over y = 2:
- (2,2) → (2,2)
- (5,2) → (5,2)
- (5,6): distance = 6 - 2 = 4 → below: 2 - 4 = -2 → (5,-2) → matches R

Yes!

So the single transformation mapping P onto R is:

> A reflection in the line y = 2

---

✔ Final Answers:

#### a) Rotate P by 180° about (1,2):

- A(2,2) → (0,2)
- B(5,2) → (-3,2)
- C(5,6) → (-3,-2)

Label this new triangle Q.

#### b) Reflect Q in the line x = 1:

- (0,2) → (2,2)
- (-3,2) → (5,2)
- (-3,-2) → (5,-2)

Label this new triangle R.

#### c) Single transformation mapping P to R:

> Reflection in the line y = 2

---

🔍 Why does this work?

Even though we went through two steps (rotation then reflection), the net effect is equivalent to a single reflection over y = 2.

You can think of it as:
- Rotating 180° about (1,2) moves P to Q
- Then reflecting Q over x=1 gives R
- But geometrically, the overall result is the same as reflecting P over y=2

This shows how combinations of transformations can sometimes be simplified.

---

✔ Final Answer for Part c:

> The single transformation that maps shape P onto R is a reflection in the line y = 2.