Scale Coordinates from the Origin
A dilation centered at the origin multiplies both coordinates by the same scale factor.
About 20 minutes, plus any extended practice
Supplied materials
- Paper or a browser scratch area; a calculator is optional unless rounding is requested.
- Supplied teaching case: Dilate (2,3) by scale factor 3 about the origin. Candidate answers: (A) (0,1) (B) (2,9) (C) (5,6) (D) (6,9).
- Independent check inputs: Dilate (3,4) by scale factor 3 about the origin. Candidate answers: (A) (1,1) (B) (3,12) (C) (6,7) (D) (9,12).
Learn the idea
Dilations change lengths by a common factor and preserve shape. Right-triangle relationships connect squared perpendicular lengths; their converse tests whether supplied side lengths support a right angle, including in a three-dimensional diagonal model. A dilation centered at the origin multiplies both coordinates by the same scale factor. The image stays on the original origin ray while its distance changes multiplicatively.
Here is the method on the supplied teaching case. Multiply both coordinates by 3 because the dilation center is the origin. The point moves along its origin ray to triple its original distance.
The center matters: multiplying coordinates directly assumes center (0,0). A dilation with factor three adds no fixed coordinate increment; it also scales lengths without preserving congruence unless the factor is one.
- Write or sketch the supplied teaching case, labeling its quantities and restrictions.
- Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
- Solve the independent check with its changed inputs before revealing feedback.
- Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
See the method in action
Teaching case: Dilate (2,3) by scale factor 3 about the origin. Candidate answers: (A) (0,1) (B) (2,9) (C) (5,6) (D) (6,9). Solution: Multiply both coordinates by 3 because the dilation center is the origin. The point moves along its origin ray to triple its original distance. The final answer is (6,9). The center matters: multiplying coordinates directly assumes center (0,0). A dilation with factor three adds no fixed coordinate increment; it also scales lengths without preserving congruence unless the factor is one.
Check your reasoning
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Helpful tips
- A dilation centered at the origin multiplies both coordinates by the same scale factor.
- The center matters: multiplying coordinates directly assumes center (0,0).
See the answer and explanation
(9,12)
Multiply both coordinates by 3 because the dilation center is the origin. The point moves along its origin ray to triple its original distance.
Take it further
Changed-input task: Dilate (9,10) by scale factor 3 about the origin. Candidate answers: (A) (12,13) (B) (27,30) (C) (3,3) (D) (9,30). Show the mathematical reason, not only the selected answer. Expected result: (27,30). Multiply both coordinates by 3 because the dilation center is the origin. The point moves along its origin ray to triple its original distance.
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