Grade 10 · Verify a distance-preserving rotation · 731 of 750
Verify a distance-preserving rotation · practice 80
Rotate (4.5, 0) about the origin by -180°. Find the image and its distance from the origin. Follow the calculation, then test the quantities in the animated example.
Try the question, use a hint when you need one, and compare your reasoning with the worked solution. The animation starts with this chapter’s values; changing its controls explores a new case.
01 · Make a prediction
Your practice question
Rotate (4.5, 0) about the origin by -180°. Find the image and its distance from the origin.
Use this scratch space or work on paper. Your notes stay on this page and clear when you leave. Answers are for self-checking; they are not automatically graded.
A starting point
Positive angles turn counterclockwise; use cosine for the horizontal coordinate and sine for the vertical coordinate.
Work through the reasoning
Step 1
Translate the givens
The initial point lies on a circle of radius 4.5; the signed angle is -180°.
Step 2
Calculate with the model
x = 4.5 cos(-180°) ≈ -4.5; y = 4.5 sin(-180°) ≈ 0.
Step 3
Check and interpret
Unrounded x² + y² = 4.5²(cos²(-180°) + sin²(-180°)) = 20.25. The distance is therefore √20.25 = 4.5.
The answer
Image = (4.5 cos(-180°), 4.5 sin(-180°)) ≈ (-4.5, 0); distance = 4.5.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
Use Play, Pause, and the timeline to inspect the construction. Reset restores the question’s original settings. The displayed assumptions describe where this model applies.
Watch the relationship
PausedHD animation studio
From experiment to screen.
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The mathematical idea
A rotation around the origin changes a point's direction while preserving its distance from the center. Positive angles turn counterclockwise and negative angles clockwise. At a quarter turn, coordinate swapping and sign changes are special cases of the general sine-cosine rule. Starting quantities: Distance from origin = 4.5; Final signed angle = -180.
(r,0) ↦ (r cos θ, r sin θ)
03 · Reflect and transfer
Explain what changes and why.
Why does a rotation preserve distance even when both coordinate values change?
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.