Instructions: Select the correct answer from each drop-down menu. ∆ABC rotates around point D to create ∆A′B′C′. Based on t

Question

20 days ago

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Instructions: Select the correct answer from each drop-down menu. ∆ABC rotates around point D to create ∆A′B′C′. Based on t

he 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.

Mathematics

2 answers:

Zina [9.1K] · Answer 1 · 2026-03-08T01:31:01+03:00

Zina [9.1K]20 days ago

7 0

The solution for the first part is (4,1).
For the second part, the answer is (11,2).

tester [8.8K] · Answer 2 · 2026-03-08T01:33:53+03:00

Responses

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

2) The coordinates of A″ are (11,2).

Solution

1) Triangle ∆ABC rotates by an angle of 180° counterclockwise around point D, forming triangle ∆A′B′C′.

The coordinates for A are (4,9)=(xa,ya) hence xa=4, ya=9.

The coordinates of A regarding the pivot D(4,5)=(xd,yd) result in xd=4, yd=5. Thus, the transformed coordinates become (xa-xd, ya-yd)=(4-4,9-5)=(0,4) leading to (x,y) where x=0, y=4.

The transformed coordinates A'=(x',y') concerning point D are calculated as follows:

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

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

Concerning the origin coordinates, A'=(xa', ya') yields:

xa'=x'+xd→xa'=0+4→xa'=4.

ya'=y'+yd→ya'=-4+5→ya'=1.

Therefore, A' coordinates are (xa',ya')=(4,1).

2) Triangle ∆A′B′C′ rotates 90° counterclockwise around point E(7, 5) to form triangle ∆A″B″C″.

Coordinates of A' are (4,1)=(xa',ya') hence xa'=4, ya'=1.

With respect to E(7,5)=(xe,ye) meaning xe=7, ye=5, the coordinates modify to (xa'-xe, ya'-ye)=(4-7,1-5)=(-3,-4) as (x,y) leads to x=-3, y=-4.

Coordinates for A''=(x'',y'') regarding point E are calculated as:

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

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

Final coordinates for A''=(xa'', ya'') concerning the origin yield:

xa''=x''+xe→xa''=4+7→xa''=11.

ya''=y''+ye→ya''=-3+5→ya''=2.

Consequently, A'' coordinates are (xa'',ya'')=(11,2).

﻿﻿﻿﻿﻿Instructions: Select the correct answer from each drop-down menu. ∆ABC rotates around point D to create ∆A′B′C′. Based on t

Instructions: Select the correct answer from each drop-down menu. ∆ABC rotates around point D to create ∆A′B′C′. Based on t