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Question

A coordinate plane shows two triangles of the same shape in quadrants 1 and 2. The horizontal axis ranges from negative 5 to 3 in unit increments. The vertical axis ranges from 0 to 7 in unit increments. Three points labeled upper A upper B upper C form a solid line triangle. The coordinates are upper A at left parenthesis negative 3 comma 5 right parenthesis, upper B at left parenthesis negative 5 comma 1 right parenthesis, and upper C at left parenthesis negative 1 comma 1 right parenthesis. The triangle shows an angle upper B of 63.4 degrees between upper A B and upper B C. A point upper D is marked at left parenthesis 0 comma 2 right parenthesis. Three points labeled upper E upper F upper G form a dotted line triangle. The coordinates are upper E at left parenthesis negative 1 comma 3 right parenthesis, upper F at left parenthesis 3 comma 5 right parenthesis, and upper G at left parenthesis negative 1 comma 7 right parenthesis. The triangle forms an angle upper E of 63.4 degrees between upper E F and upper E G, an angle G of 63.4 degrees between upper E G and upper G F, and an angle upper F of 53.2 degrees between upper G F and upper E F.

Find the measurement of angle A
in the rotation of ΔABC
90°
clockwise around point D
.

(1 point)
____degrees

Answer 1

To find the measurement of angle A in the rotation of triangle ΔABC 90° clockwise around point D, we will first identify the relevant components of the problem.

1. Original Triangle ΔABC:

   • Point A is at (-3, 5)
   • Point B is at (-5, 1)
   • Point C is at (-1, 1)
   • Angle B is given as 63.4°.

2. Point D:

   • Point D is at (0, 2), which serves as the center of rotation.

3. Understanding the Rotation:

   • When rotating a point (x, y) 90° clockwise around another point (h, k), the new coordinates (x', y') can be calculated using the transformation: \[ x' = h + (y - k) \] \[ y' = k - (x - h) \]

4. Calculating the New Positions:

   • Rotate Point A (-3, 5): \[ x' = 0 + (5 - 2) = 3 \] \[ y' = 2 - (-3 - 0) = 2 + 3 = 5 \] So, the new position of A' is (3, 5).

   • Rotate Point B (-5, 1): \[ x' = 0 + (1 - 2) = 0 - 1 = -1 \] \[ y' = 2 - (-5 - 0) = 2 + 5 = 7 \] So, the new position of B' is (-1, 7).

   • Rotate Point C (-1, 1): \[ x' = 0 + (1 - 2) = 0 - 1 = -1 \] \[ y' = 2 - (-1 - 0) = 2 + 1 = 3 \] So, the new position of C' is (-1, 3).

5. Finding the New Angles:

   • Triangle ΔA'B'C' is formed with the new coordinates:

     • A' (3, 5)
     • B' (-1, 7)
     • C' (-1, 3)

   • To find angle A' (which corresponds to angle A), we note that the original triangle ΔABC has angle A, which remains the same since rotation being rigid preserves angles.

Since the notation says that angle A was originally at a certain angle before rotation and not affected by the rotation itself, it can be stated that Angle A remains 90° - angle B of the original triangle.

Thus, the measurement of angle A, when rotated 90° clockwise around point D, would remain: \[ \boxed{26.6} \text{ degrees} \] (calculated as \(90° - 63.4°\)).