Question

Triangle PQR is dilated by a scale factor of 1.5 to form triangle P′Q′R′. This triangle is then dilated by a scale factor of 2 to form triangle P″Q″R″. If vertex P is at (3, -4) and vertex Q is at (5, -1), what are the coordinates of vertices P″ and Q″?
P″(4.5, -6) and Q″(7.5, 1.5)
P″(4.5, -6) and Q″(7.5, -1.5)
P″(-9, 12) and Q″(15, -3)
P″(9, -12) and Q″(15, -3)
P"(-9, -12) and Q"(-15, -3)

Answer 1

Answer: P″(9, -12) and Q″(15, -3)

Step-by-step explanation:

When a figure with coordinate (x,y) dilated by scale factor k , then the coordinates of the image is given by :-

$(x,y)\to(kx,ky)$

Given : The coordinates of vertex P is (3, -4) and Q is (5, -1) .

Triangle PQR is dilated by a scale factor of 1.5 to form triangle P′Q′R′.

Then, the co-ordinates of P' and Q' are given by:-

$P(3,-4)\to P'(1.5\cdot3,\ 1.5\cdot-4)=P'(4.5,\ -6)$

$Q(5,-1)\to Q'(1.5\cdot5,\ 1.5\cdot-1)=P'(7.5,\ -1.5)$

Triangle P′Q′R′ is then dilated by a scale factor of 2 to form triangle P″Q″R″.

Then, the co-ordinates of P" and Q" are given by:-

$P"(4.5,\ -6)\to P"(2\cdot4.5,\ 2\cdot-6)=P"(9,\ -12)$

$Q'(7.5,\ -1.5)\to Q'(2\cdot7.5,\ 2\cdot-1.5)=P'(15,\ -3)$

Hence, the coordinates of vertices P″ and Q″ are P″(9, -12) and Q″(15, -3).

Answer 2

Answer:  The correct option is (D) P″(9, -12) and Q″(15, -3).

Step-by-step explanation:  Given that triangle PQR is dilated by a scale factor of 1.5 to form triangle P′Q′R′. This triangle is then dilated by a scale factor of 2 to form triangle P″Q″R″.

The co-ordinates of vertices P and Q are (3, -4) and (5, -1) respectively.

We are to find the co-ordinates of the vertices P″ and Q″.

Case I :  ΔPQR dilated to ΔP'Q'R'

The co-ordinates of P' and Q' are given by

$P'(3\times 1.5, -4\times 1.5)=P'(4.5, -6),\\\\Q'(5\times 1.5, -1\times 1.5)=Q'(7.5, -1.5).$

Case II :  ΔP'Q'R' dilated to ΔP''Q''R''

The co-ordinates of P'' and Q'' are given by

$P''(4.5\times 2, -6\times 2)=P'(9, -12),\\\\Q'(7.5\times 2, -1.5\times 2)=Q'(15, -3).$

Thus, the co-ordinates of the vertices P'' and Q'' are (9, -12) and (15, -3).

Option (D) is CORRECT.