(a)
(i) Enlarge triangle T by scale factor 3, centre (0,2).
(ii)(a) Rotate triangle T about (4,2) by 90° clockwise. Label the image P.
(ii)(b) Reflect triangle T in the line x+y=6. Label the image Q.
(ii)(c) Describe fully the single transformation that maps triangle P onto triangle Q.
(b)
The diagram shows triangle OHK, where O is the origin. The position vector of H is a and the position vector of K is b. Z is the point on HK such that HZ:ZK=2:5.
Find the position vector of Z, in terms of a and b. Give your answer in its simplest form.
▶️ Answer/Explanation
(a)(i) Triangle at (3,-1), (9,-1), (9,2)
Each point’s distance from center (0,2) is multiplied by 3.
(a)(ii)(a) Triangle at (3,3), (4,3), (3,5)
Rotated 90° clockwise about (4,2), changing coordinates accordingly.
(a)(ii)(b) Triangle at (4,3), (5,3), (5,5)
Reflected over x+y=6, swapping x and y coordinates relative to the line.
(a)(ii)(c) Reflection in x=4
Comparing P and Q, they are mirror images across the vertical line x=4.
(b) (5/7)a + (2/7)b
Z divides HK in 2:5 ratio, so its position vector is a weighted average of a and b.
(a) Describe fully the single transformation that maps triangle \( A \) onto triangle \( B \).
(b) Draw the image of triangle \( A \) after
(i) a reflection in the line \( y = 1 \)
(ii) a translation by the vector \(\begin{pmatrix} 5 \\ -7 \end{pmatrix}\)
(iii) an enlargement, scale factor 2, centre (\(-4\), \(5\)).
▶️ Answer/Explanation
(a) Rotation 90° anticlockwise about the point (2, 7)
Triangle A is rotated 90 degrees counter-clockwise around the center point (2,7) to match the position of triangle B.
(b)(i) Image at (-4, -1), (-3, -1), (-4, -4)
Each point of triangle A is reflected across the horizontal line y=1, flipping the triangle vertically while maintaining equal distances from the mirror line.
(b)(ii) Image at (2, -4), (1, -4), (1, -1)
Every point of triangle A is moved right 5 units and down 7 units according to the translation vector (5,-7).
(b)(iii) Image at (-4, 7), (-4, 1), (-2, 1)
Triangle A is enlarged by scale factor 2 from the center point (-4,5), doubling all distances from this point while maintaining the shape.