Question

Instructions: Select the correct answer from each drop-down menu. ∆ABC rotates around point D to create ∆A′B′C′. Based on the position of, shown in the figure, the coordinates of A′ are. If ∆A′B′C′ rotates 90° counterclockwise around point E(7, 5) to form triangle ∆A″B″C″, the coordinates of A″ are.

Answer 1

The answer to the first part is (4,1).
The answer to the second part is (11,2).

Answer 2

Answers

1) The coordinates of A' are (4,1).

2) The coordinates of A″ are (11,2).

Solution

1) The ∆ABC rotates an angle A=180° counterclocwise around point D to create ∆A′B′C′

The coordinates of A are (4,9)=(xa,ya)→xa=4, ya=9

The coordinates of A with respect the point of rotation D(4,5)=(xd,yd)→xd=4, yd=5 are (xa-xd, ya-yd)=(4-4,9-5)=(0,4)=(x,y)→x=0, y=4

The coordinates of A'=(x',y') with respect the point of rotation D are:

x'= x cos A - y sin A

x' = 0 cos 180° - 4 sin 180°

x'= 0 (-1) - 4 (0)

x'=0-0

x'=0

y' = y cos A + x sin A

y' = 4 cos 180° + 0 sin 180°

y' = 4 (-1) + 0 (0)

y' = -4+0

y'= -4

(x',y')=(0,-4)

The coordinates of A'=(xa', ya') with respect the origin are:

xa'=x'+xd→xa'=0+4→xa'=4

ya'=y'+yd→ya'=-4+5→ya'=1

The coordinates of A' are (xa',ya')=(4,1)

2) The ∆A′B′C′ rotates an angle B=90° counterclockwise around point E(7, 5) to form triangle ∆A″B″C″

The coordinates of A' are (4,1)=(xa',ya')→xa'=4, ya'=1

The coordinates of A' with respect the point of rotation E(7,5)=(xe,ye)→xe=7, ye=5 are (xa'-xe, ya'-ye)=(4-7,1-5)=(-3,-4)=(x,y)→x=-3, y=-4

The coordinates of A''=(x'',y'') with respect the point of rotation E are:

x''= x cos B - y sin B

x'' = -3 cos 90° - (-4) sin 90°

x''= -3(0) + 4 (1)

x''= -0+4

x''=4

y'' = y cos B + x sin B

y'' = -4 cos 90° + (-3) sin 90°

y'' = -4 (0) - 3 (1)

y'' = -0-3

y''= -3

(x'',y'')=(4,-3)

The coordinates of A''=(xa'', ya'') with respect the origin are:

xa''=x''+xe→xa''=4+7→xa''=11

ya''=y''+ye→ya''=-3+5→ya''=2

The coordinates of A'' are (xa'',ya'')=(11,2)