1. **Problem 1: Draw and reflect triangle ABC** - Given vertices: A(-5,-2), B(-3,-4), C(-1,-2). - Reflect triangle ABC in the line $y=0$ (the x-axis). - Then reflect the image triangle A'B'C' in the y-axis. 2. **Reflection formulas:** - Reflection in $y=0$: $(x,y) \to (x,-y)$. - Reflection in y-axis: $(x,y) \to (-x,y)$. 3. **Reflect ABC in $y=0$:** - $A(-5,-2) \to A'(-5,2)$ - $B(-3,-4) \to B'(-3,4)$ - $C(-1,-2) \to C'(-1,2)$ 4. **Reflect A'B'C' in y-axis:** - $A'(-5,2) \to A''(5,2)$ - $B'(-3,4) \to B''(3,4)$ - $C'(-1,2) \to C''(1,2)$ (correcting given $C''$ coordinates) --- 5. **Problem 2: Enlargement scale factor and length** - Given $AB=3$ cm, $A'B'=6$ cm. - Scale factor $k = \frac{A'B'}{AB} = \frac{6}{3} = 2$. 6. **Find $B'C'$ given $BC=4.5$ cm:** - $B'C' = k \times BC = 2 \times 4.5 = 9$ cm. --- 7. **Problem 3: Area scale factor of enlargement** - Original area = 4 cm$^2$, image area = 64 cm$^2$. - Area scale factor $= \frac{\text{image area}}{\text{original area}} = \frac{64}{4} = 16$. --- 8. **Problem 4: Transformation of points P, A, B, C, O** - Given points and images: - $P(2,3) \to P'(y,x) = (3,2)$ - $A(4,3) \to A'(3,4)$ - $B(4,4) \to B'(4,4)$ - $C(6,4) \to C'(4,6)$ - $O(7,2) \to O'(2,7)$ 9. **Describe transformation:** - The transformation swaps coordinates: $(x,y) \to (y,x)$. - This is a reflection in the line $y=x$. --- 10. **Problem 5: Enlargement of triangle P** - Given scale factors from ratios: - $k = \frac{A'B'}{AB} = \frac{2.5}{1.2} = \frac{25}{12} \approx 2.08$ - $k = \frac{B'C'}{BC} = \frac{12}{5} = 2.4$ - These differ slightly; likely approximate or measurement error. - The centre of enlargement is the point where lines joining corresponding vertices intersect (not given explicitly). --- 11. **Problem 6: Enlargement of triangle ABC to A'B'C'** - Given: - $A(1,1), B(4,5), C(2,4)$ - $A'(2,0), B'(8,8), C'(4,6)$ 12. **Check if A'B'C' is enlargement of ABC:** - Calculate lengths: - $AB = \sqrt{(4-1)^2 + (5-1)^2} = \sqrt{9 + 16} = 5$ - $A'B' = \sqrt{(8-2)^2 + (8-0)^2} = \sqrt{36 + 64} = 10$ - Scale factor $k = \frac{A'B'}{AB} = \frac{10}{5} = 2$. 13. **Centre of enlargement:** - Draw lines $AA'$, $BB'$, $CC'$ and find their intersection. - Using coordinates, solve equations of lines: - $AA'$ passes through $(1,1)$ and $(2,0)$ - $BB'$ passes through $(4,5)$ and $(8,8)$ - $CC'$ passes through $(2,4)$ and $(4,6)$ - Intersection point is centre of enlargement. 14. **Calculate centre:** - Line $AA'$ slope: $m = \frac{0-1}{2-1} = -1$; equation: $y-1 = -1(x-1) \Rightarrow y = -x + 2$ - Line $BB'$ slope: $m = \frac{8-5}{8-4} = \frac{3}{4}$; equation: $y-5 = \frac{3}{4}(x-4) \Rightarrow y = \frac{3}{4}x + 2$ - Solve for intersection: $$-x + 2 = \frac{3}{4}x + 2$$ $$-x = \frac{3}{4}x$$ $$-x - \frac{3}{4}x = 0$$ $$-\frac{7}{4}x = 0 \Rightarrow x=0$$ - Substitute $x=0$ into $y = -x + 2$: $y=2$ - Centre of enlargement is $(0,2)$. 15. **Scale factor confirmed as 2.** --- 16. **Problem 7: Construct image of triangle ABC under enlargement centre (0,0) and factor $-\frac{1}{2}$** - Given points $A(1,1), B(3,3), C(2,2.5)$. - Enlargement formula: $(x,y) \to (k x, k y)$ with $k = -\frac{1}{2}$. 17. **Calculate images:** - $A' = (1 \times -\frac{1}{2}, 1 \times -\frac{1}{2}) = (-0.5, -0.5)$ - $B' = (3 \times -\frac{1}{2}, 3 \times -\frac{1}{2}) = (-1.5, -1.5)$ - $C' = (2 \times -\frac{1}{2}, 2.5 \times -\frac{1}{2}) = (-1, -1.25)$ --- 18. **Problem 8: Translation of triangle ABC by $T=(-2,3)$** - Given points $A(1,1), B(3,3), C(2,2.5)$. - Translation formula: $(x,y) \to (x + t_x, y + t_y)$ with $t_x = -2$, $t_y = 3$. 19. **Calculate images:** - $A' = (1 - 2, 1 + 3) = (-1, 4)$ - $B' = (3 - 2, 3 + 3) = (1, 6)$ - $C' = (2 - 2, 2.5 + 3) = (0, 5.5)$ --- **Final answers:** - Reflected points $A''(5,2), B''(3,4), C''(1,2)$. - Scale factor of enlargement $k=2$. - Length $B'C' = 9$ cm. - Area scale factor $=16$. - Transformation $(x,y) \to (y,x)$ is reflection in $y=x$. - Centre of enlargement for ABC to A'B'C' is $(0,2)$. - Enlargement image points with factor $-\frac{1}{2}$: $A'(-0.5,-0.5), B'(-1.5,-1.5), C'(-1,-1.25)$. - Translation image points: $A'(-1,4), B'(1,6), C'(0,5.5)$.