Math Problem Statement

Bayangan titik Q(-4,5) oleh translasi T = (-1,4) dilanjutkan dengan rotasi sejauh 90 derajat berlawanan arah jarum jam terhadap pusat O(0,0) adalah ...

Solution

The problem in the image asks for the image of the point Q(4,5)Q(-4, 5) after two transformations:

  1. Translation by T=(14)T = \left( \frac{-1}{4} \right), which seems to refer to a vector translation of (1,4)\left( -1, 4 \right).
  2. Rotation by 9090^\circ counterclockwise about the origin O(0,0)O(0, 0).

Step 1: Translation

We first translate point Q(4,5)Q(-4, 5) by the vector (1,4)(-1, 4).

  • New coordinates of QQ': Q=(41,5+4)=(5,9)Q' = (-4 - 1, 5 + 4) = (-5, 9)

Step 2: Rotation

Next, we rotate the point Q(5,9)Q'(-5, 9) counterclockwise by 9090^\circ around the origin O(0,0)O(0, 0). The transformation formula for a 9090^\circ counterclockwise rotation is:

  • New coordinates of QQ'' after rotation: Q=(y,x)Q'' = (-y, x) Applying this to (5,9)(-5, 9): Q=(9,5)Q'' = (-9, -5)

Final Answer:

The image of the point after translation and rotation is (9,5)(-9, -5).


Would you like further details or explanations? Here are some related questions to practice:

  1. How would the result change if the rotation was 9090^\circ clockwise?
  2. What is the general formula for rotating a point by 180180^\circ?
  3. How do you apply a translation using different vectors?
  4. What are the coordinates of the point if the rotation was around a different center?
  5. What happens if the translation vector is reversed?

Tip: Always apply translation first, followed by rotation, unless the problem specifies otherwise.

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Math Problem Analysis

Mathematical Concepts

Translation
Rotation
Coordinate Geometry

Formulas

Translation formula: (x', y') = (x + dx, y + dy)
Rotation formula for 90 degrees counterclockwise: (x', y') = (-y, x)

Theorems

Rotation Theorem

Suitable Grade Level

Grades 8-10