Circle Blueprint SlidesCircle Blueprints Proving All Circles are Similar Intervention Series | Geometry Our Objective Students will prove that any two circles are similar by identifying the sequence of transformations that maps one onto the other. Colorado Standard: HS.G-C.A.1: Prove that all circles are similar. Review: What is Similarity? Rigid Motions • Translation (Slide) • Rotation (Turn) • Reflection (Flip) Keeps same size & shape Dilation • Scaling up or down • Uses a Scale Factor (k) Keeps same shape, different size Similarity = Sequence of Rigid Motions + Dilation Circle Components Circle A r₁ Circle B r₂ Every circle is defined by just two things: 1. Center Point (x, y) 2. Radius (r) To prove similarity, we must move the Center and scale the Radius. The Blueprint for Success 1 Translate Move the center of Circle 1 to the center of Circle 2. 2 Dilate Scale the radius of Circle 1 to match the radius of Circle 2. Scale Factor Formula \[ k = \frac{r_{target}}{r_{source}} \] "New Radius divided by Old Radius" Guided Practice Circle A: Center (2, 3), Radius = 4 Circle B: Center (8, -1), Radius = 12 1. Translate: Move (2, 3) to (8, -1) (x + 6, y - 4) 2. Dilate: Radius 4 to Radius 12 k = 12 / 4 = 3 Visualizing the Shift... Why ALL circles? Because any center can be translated to another center... AND Any radius can be scaled to another radius using a scale factor! Since these transformations exist for any two circles, they are all similar.
Similarity Squad WorksheetSimilarity Squad Worksheet Mission: Proving All Circles are Similar Agent: Date: 1. Blueprint Vocabulary Draw a line to connect the term to its correct definition. Translation Dilation Scale Factor (k) Changes the size to match the radii. The ratio of the new radius to the old radius. Moves one center to the other location. 2. Guided Proof: Circle A to Circle B Source: Circle A Center: (1, 2) Radius: 3 units Target: Circle B Center: (5, -1) Radius: 12 units 1 Identify the Translation Calculate the horizontal and vertical shift needed to move (1, 2) to (5, -1). HORIZONTAL SHIFT (x) VERTICAL SHIFT (y) 2 Calculate the Scale Factor (k) Divide the Target radius by the Source radius. New Radius / Old Radius = Scale Factor (k) 3. The Similarity Proof Prove Circle C is similar to Circle D by providing the specific sequence of transformations. Circle C (Source) Center: (0, 0) Radius: 5 Circle D (Target) Center: (-3, 7) Radius: 2.5 Step 1: Transformation Rule for Translation Step 2: Scale Factor Calculation Formal Similarity Statement Circle C is similar to Circle D because there exists a sequence of transformations, specifically a translation of ___________________ units and a dilation with a scale factor of ___________________, that maps Circle C exactly onto Circle D.
Intervention Roadmap GuideIntervention Roadmap Circle Similarity Proofs | Teacher Guide Tier 2 Intervention Prerequisite Skills Identify center/radius of a circle. Apply translation rules \((x+h, y+k)\). Understand dilation and scale factor. Common Pitfalls Scale Factor Reversal Students often do \(r_{source} / r_{target}\) instead of \(target/source\). Emphasize "New over Old". Radius vs Diameter Ensure students aren't mixing up diameter measurements with radius when dilating. Facilitation Script Launch (3 mins) "Look at these two circles on the slide. They look like the same shape, just different sizes. In Geometry, what do we call shapes that are the same shape but different sizes?" (Wait for: Similar) Investigation (10 mins) "To prove they are similar, we need to map one onto the other. If I move the center of Circle A to Circle B, did the shape change? No, just its location. Now, how do I make Circle A grow or shrink to match Circle B's size?" Conclusion (2 mins) "Since we can ALWAYS move a center and ALWAYS scale a radius, we can prove that ANY two circles in the universe are similar." Progress Monitor Student NameTranslation CheckDilation CheckProof Logic□□□□□□□□□□□□ Quick Assessment: Exit Ticket "Circle X has radius 4. Circle Y has radius 10. What scale factor maps Circle X onto Circle Y?" A) k = 0.4 B) k = 2.5 C) k = 6 D) k = 14 Concept Check: "True or False: If two circles have the same center but different radii, they are still similar." TRUE FALSE
Circle Similarity Exit TicketBlueprint Check Exit Ticket: Circle Similarity Name: Date: 1 Circle P has a radius of 6 units. Circle Q has a radius of 15 units. What scale factor (\(k\)) would be used in a dilation to map Circle P onto Circle Q? \(k = 0.4\) \(k = 2.5\) \(k = 9\) \(k = 21\) 2 Briefly explain: Why can we say that ALL circles are similar, but we CANNOT say that all rectangles are similar? 3 State the two types of transformations needed to prove any two circles are similar.
