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)

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

[tex]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).[/tex]

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

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

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

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

Option (D) is CORRECT.

JeanaShupp JeanaShupp · Answer 2 · 2019-11-02T07:50:18+01:00

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 :-

[tex](x,y)\to(kx,ky)[/tex]

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:-

[tex]P(3,-4)\to P'(1.5\cdot3,\ 1.5\cdot-4)=P'(4.5,\ -6)[/tex]

[tex]Q(5,-1)\to Q'(1.5\cdot5,\ 1.5\cdot-1)=P'(7.5,\ -1.5)[/tex]

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:-

[tex]P"(4.5,\ -6)\to P"(2\cdot4.5,\ 2\cdot-6)=P"(9,\ -12)[/tex]

[tex]Q'(7.5,\ -1.5)\to Q'(2\cdot7.5,\ 2\cdot-1.5)=P'(15,\ -3)[/tex]

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