Question

△abc rotates around point d to create △abc.
based on the position of \overline{bc}, shown in the figure, the coordinates of a are
if △abc rotates 90° counterclockwise around point e(7,5) to form triangle △abc, the coordinates of a are

Explanation:

Step1: Determine coordinates of original points

First, find coordinates of \( B \), \( C \), \( B' \), \( C' \). From graph: \( B(4,6) \), \( C(3,6) \), \( B'(4,4) \), \( C'(6,4) \). The rotation from \( BC \) to \( B'C' \): \( BC \) is horizontal (y=6, x from 3 to 4). \( B'C' \) is horizontal (y=4, x from 4 to 6). The translation/rotation: vector from \( B \) to \( B' \): \( (4 - 4, 4 - 6)=(0, -2) \), from \( C \) to \( C' \): \( (6 - 3, 4 - 6)=(3, -2) \). Wait, better to see the rotation center \( D \). From \( B(4,6) \) to \( B'(4,4) \): vertical shift down 2, \( C(3,6) \) to \( C'(6,4) \): right 3, down 2. Wait, maybe rotation 90? No, \( BC \) length is 1 (from x=3 to 4, y=6), \( B'C' \) length is 2? Wait no, \( B(4,6) \), \( C(3,6) \): length \( |4 - 3| = 1 \). \( B'(4,4) \), \( C'(6,4) \): length \( |6 - 4| = 2 \). Wait, maybe I misread. Wait, original \( \triangle ABC \): \( B \) is (4,6), \( C \) is (3,6), \( A \) is (4,9) (since from B(4,6) up 3 to A(4,9)). Now, \( B' \) is (4,4), \( C' \) is (6,4). So vector \( B \) to \( B' \): (0, -2), \( C \) to \( C' \): (3, -2). Wait, maybe the rotation is 90 degrees? Wait, no, let's find \( A \)'s coordinates. \( A \) is (4,9). Let's see the rotation: from \( B(4,6) \) to \( B'(4,4) \): down 2, \( C(3,6) \) to \( C'(6,4) \): right 3, down 2. Wait, maybe the rotation center \( D \) is (4,5) (since \( D \) is at (4,5) from graph). So vector from \( D \) to \( B \): (0,1), to \( B' \): (0, -1). So rotation 180? No, 180 would be (4 - (4 - 4), 5 - (6 - 5))=(4,4), which matches \( B' \). Similarly, \( C \) is (3,6): vector from \( D(4,5) \) to \( C \) is (-1,1), so 180 rotation: \( D + (-(-1), -1)=(4 + 1, 5 - 1)=(5,4) \)? No, \( C' \) is (6,4). Wait, maybe my initial coordinates are wrong. Let's re-express:

Looking at the graph: x-axis from -2 to 7, y-axis from -1 to 9. \( B \) is at (4,6) (x=4, y=6), \( C \) is at (3,6) (x=3, y=6), \( A \) is at (4,9) (x=4, y=9). \( B' \) is at (4,4) (x=4, y=4), \( C' \) is at (6,4) (x=6, y=4). So the vector from \( B \) to \( B' \) is (0, -2), from \( C \) to \( C' \) is (3, -2). Wait, the length of \( BC \) is 1 (horizontal), \( B'C' \) is 2 (horizontal). No, that can't be. Wait, maybe \( C \) is (3,6), \( B \) is (4,6), so \( BC \) is horizontal left to right? No, \( C \) is (3,6), \( B \) is (4,6): so \( C \) is left of \( B \), \( BC \) vector is (1,0). \( B' \) is (4,4), \( C' \) is (6,4): \( B'C' \) vector is (2,0). Wait, maybe the rotation is a translation? No, the problem says "rotates around point D". So center \( D \). Let's find \( D \)'s coordinates: from graph, \( D \) is at (4,5) (between \( B(4,6) \) and \( B'(4,4) \), y=5, x=4). So vector from \( D \) to \( B \): (0,1), to \( B' \): (0, -1). So rotation 180 degrees? Because rotating 180 around \( D(4,5) \): a point \( (x,y) \) rotated 180 around \( (h,k) \) is \( (2h - x, 2k - y) \). So for \( B(4,6) \): \( 2*4 - 4 = 4 \), \( 2*5 - 6 = 4 \), so \( B'(4,4) \), correct. For \( C(3,6) \): \( 2*4 - 3 = 5 \)? Wait no, \( C' \) is (6,4). Wait, \( 2*4 - 3 = 5 \), \( 2*5 - 6 = 4 \), but \( C' \) is (6,4). So my \( C \) coordinate is wrong. Wait, maybe \( C \) is (3,6), \( B \) is (4,6), \( D \) is (4,5). Then \( C \) rotated 180 around \( D \): \( (2*4 - 3, 2*5 - 6)=(5,4) \), but \( C' \) is (6,4). So maybe \( C \) is (2,6)? No, graph shows \( C \) at x=3. Wait, maybe the rotation is 90 degrees clockwise? Rotation 90 clockwise around \( D(4,5) \): formula \( (x,y) \to (h + (y - k), k - (x - h)) \). For \( B(4,6) \): \( h=4, k=5 \), so \( (4 + (6 - 5), 5 - (4 - 4))=(5,5) \), but \( B' \) is (4,4). Not matching. Rotation 90 counterclockwise: \( (h - (y - k), k + (x - h)) \). For \( B(4,6) \): \( (4 - (6 - 5), 5 + (4 - 4))=(3,5) \), no. Wait, maybe the translation is down 2 and right 0 for \( B \), down 2 and right 3 for \( C \). So \( A \) is (4,9). So \( A \) rotated: down 2 and right 0? No, \( B \) went from (4,6) to (4,4) (down 2), \( C \) from (3,6) to (6,4) (right 3, down 2). So the vector from \( A \) to \( A' \) should be same as from \( C \) to \( C' \) minus from \( B \) to \( B' \)? Wait, \( A \) is (4,9), \( B \) is (4,6), so \( AB \) vector is (0,3) (vertical up). So \( A' \) should have \( B'A' \) vector same as \( AB \) vector, but rotated. Wait, \( AB \) is vertical (from \( B(4,6) \) to \( A(4,9) \), vector (0,3)). So \( B' \) is (4,4), so \( A' \) should be (4,4 + 3)=(4,7)? Wait, no, because \( C \) to \( C' \) is (3, -2), \( B \) to \( B' \) is (0, -2). So the difference in x for \( C \) and \( B \) is -1 (3 - 4 = -1), for \( C' \) and \( B' \) is 2 (6 - 4 = 2). So the scale factor? Wait, maybe the rotation is a 90 degree rotation. Wait, let's check the options: (4,1), (4,7), (6,1), (6,7). Let's see the original \( A(4,9) \). Rotating around \( D(4,5) \): distance from \( D \) to \( A \) is \( 9 - 5 = 4 \) (vertical). Rotating 180 degrees around \( D \): \( A' \) would be \( (4, 5 - 4)=(4,1) \)? No, 180 rotation: \( (2*4 - 4, 2*5 - 9)=(4,1) \). Wait, but \( B \) rotated 180 around \( D(4,5) \): \( (2*4 - 4, 2*5 - 6)=(4,4) \), correct. \( C \) rotated 180 around \( D(4,5) \): \( (2*4 - 3, 2*5 - 6)=(5,4) \), but \( C' \) is (6,4). So my \( C \) coordinate is wrong. Wait, maybe \( C \) is (2,6), \( B \) is (4,6). Then \( C(2,6) \) rotated 180 around \( D(4,5) \): \( (2*4 - 2, 2*5 - 6)=(6,4) \), correct! \( B(4,6) \) rotated 180 around \( D(4,5) \): \( (2*4 - 4, 2*5 - 6)=(4,4) \), correct. Ah! So \( C \) is (2,6), \( B \) is (4,6), so \( BC \) length is 2 (from x=2 to 4, y=6), \( B'C' \) length is 2 (from x=4 to 6, y=4), correct. So \( C(2,6) \), \( B(4,6) \), \( A(4,9) \) (since \( AB \) is vertical from \( B(4,6) \) up to \( A(4,9) \)). Now, rotate \( A(4,9) \) 180 degrees around \( D(4,5) \): formula \( (x', y') = (2h - x, 2k - y) \), where \( (h,k)=(4,5) \), \( (x,y)=(4,9) \). So \( x' = 2*4 - 4 = 4 \), \( y' = 2*5 - 9 = 10 - 9 = 1 \)? No, \( 2*5 - 9 = 1 \)? Wait, \( 5*2 = 10 \), \( 10 - 9 = 1 \). But option (4,1) is there. Wait, but \( B \) rotated 180: \( (4,6) \to (4,4) \) (since \( 2*5 - 6 = 4 \)), correct. \( C(2,6) \to (6,4) \) ( \( 2*4 - 2 = 6 \), \( 2*5 - 6 = 4 \) ), correct. So \( A(4,9) \) rotated 180 around \( D(4,5) \): \( (4, 2*5 - 9)=(4,1) \)? Wait, no, \( 2k - y = 10 - 9 = 1 \), yes. But wait, the options have (4,7) too. Wait, maybe rotation 90 degrees clockwise? Rotation 90 clockwise around \( D(4,5) \): formula \( (x,y) \to (h + (y - k), k - (x - h)) \). For \( A(4,9) \): \( h=4, k=5 \), so \( (4 + (9 - 5), 5 - (4 - 4))=(8,5) \), not in options. Rotation 90 counterclockwise: \( (h - (y - k), k + (x - h))=(4 - (9 - 5), 5 + (4 - 4))=(0,5) \), no. Wait, maybe the translation is down 2 and left 0 for \( B \), down 2 and right 2 for \( C \). Wait, \( C(2,6) \) to \( C'(6,4) \): right 4? No, 6 - 2 = 4. \( B(4,6) \) to \( B'(4,4) \): down 2. No, this is confusing. Wait, the options are (4,1), (4,7), (6,1), (6,7). Let's think about the vertical line. \( A \) is (4,9), \( B \) is (4,6), so \( AB \) is vertical (x=4). After rotation, \( A' \) should have x-coordinate related to \( B' \)'s x (4) or \( C' \)'s x (6). If \( AB \) is vertical, then \( A'B' \) should also be vertical (since rotation preserves length and angle). \( B' \) is (4,4), so if \( A'B' \) is vertical, \( A' \) has x=4. So options (4,1) or (4,7). The distance from \( B \) to \( A \) is 3 (9 - 6 = 3). So distance from \( B' \) to \( A' \) should be 3. \( B' \) is (4,4). So \( A' \) could be (4,4 + 3)=(4,7) or (4,4 - 3)=(4,1). Now, check the rotation direction. The original triangle is above \( B'C' \), rotating around \( D \) (which is between \( B \) and \( B' \), y=5). So rotating down: from \( A(4,9) \) to \( A'(4,1) \) (down 8) or up? No, \( B \) went down 2 (from 6 to 4), so \( A \) should go down 2? No, \( A \) is 3 above \( B \), so \( A' \) should be 3 above \( B' \)? \( B' \) is 4, so 4 + 3 = 7. So (4,7). Let's verify: \( B(4,6) \) to \( B'(4,4) \): down 2, \( A(4,9) \) to \( A'(4,7) \): down 2, which is consistent. And \( C(3,6) \) to \( C'(6,4) \): right 3, down 2. Wait, \( C \) is (3,6), down 2 is 4, right 3 is 6, so \( C'(6,4) \), correct. So \( A(4,9) \) down 2 is (4,7). Yes, that makes sense. So the coordinates of \( A' \) are (4,7).

Step2: Confirm with rotation

Rotation around \( D \): the vector from \( B \) to \( A \) is (0,3) (up 3). So vector from \( B' \) to \( A' \) should also be (0,3) (up 3) because rotation preserves the length and direction (if it's a translation-rotation, but here it's a rotation that moves \( B \) down 2, so \( A \) moves down 2 from 9 to 7, x remains 4. So \( A'(4,7) \).

Answer:

(4, 7)