Math Problem Statement
The user is asking to rotate a triangle 90° counterclockwise about the origin, reflect it over the x-axis, and provide the final coordinates of the transformed triangle.
Solution
To solveTo solve the problem step by step this:
problem###:
Step1. Initial Points of :
From the grid:
- Identify the coordinates of (
\ triangle - - The vertices1 of are -: -1 () R (-)3
,2 .4) ) **Rotate- ( ( S(-1, \3triangle) \ R) ST ) 90° Counter-
Step clockwise2: Rotate 90° counterclockwise the about the origin
The Origin rule for a:** 90° counterclockwise rotation is:
\ The] Applying this rule: - \( rule for 90° counterclockwise rotation around the origin is: \[ (x, y) \rightarrow (-y, x)$$ R(-3, 4) \to R'(-4, -3) \) - $$S(-1, 3) \to S'(-3, -1)$$ - $$T(-2, 1) \to T'(-1, -2)$$ Thus, the rotated triangle has vertices: $$R'(-4, -3), \; S'(-3, -1), \; T'(-1, -2)$$ ### Step 3: Reflect the rotated triangle over the x-axis The rule for reflecting a point over the x-axis is: $$(x, y) Applying this \: - $$R(-3,to 4) \rightarrow R'(-4, -3)$$ - $$S ((-2, 1) \rightarrow S'(-1, -2)$$ -x, - \(y T) \(-] 1Applying this, rule: --1 $$R)'(- \4, -rightarrow T3)'( \1to R''(-, -4, 13))$$ 3. **Reflect $$\triangle R'S'T'$$ Over the x-axis:** The rule for reflection over the x-axis is: \ \[ ) - \( S'(-3, -1) \to S''(-3, 1 (x) \,) - $$y T'(-) \rightarrow1, (x -2,) \to T -''(-y1, 2)$$ Thus, the final triangle has) vertices: \[ R''(-$$ 4 Applying this,: 3), \; S'' (-3, 1), \; - T'' \( R'(-(-41, 2) \] ###, Step 4: Plot the points -3 You would) plot the triangle with the \ following vertices: - $$R''rightarrow R''(-4, 3(-4, 3))$$ - $$S''(-3, 1)$$ - $$T''(-1, 2)$$ If you'd like, \) - $$S'(-1, - I can2 create a visualization for) \ thisrightarrow transformation S''(-. Would you like me to generate1, a graph showing the original triangle 2, its)$$ rotation , - \( and T'( its1, -1) \rightarrow T'' reflection?(1, 1) \) 4. **Final Points of the Trans--- formed#### $$\triangle R Follow''-upS'' QuestionsT''$$:: ** - \( R''(-4, 3)1. How do you find the coordinates of a point \ after a rotation by 180) ° - $$S''(-1, 2)$$ about the - $$T''(1, 1)$$ Would you like a detailed graph of the transformations? Let me know! ### Questions to Expand origin Understanding: ? 21.. What happens when the What triangle is rotated changes clockwise when instead of counterclock youwise reflect? a2. How would the point over process the differ y if-axis the instead triangle of were the reflected over the y-axis instead x? 3-axis? 3. What is. the Can rule for you a general 180° rotationize about the the origin rotation? 4. How rule would for we an find the coordinates if the triangle were dilated instead of rotated angle? 5 other than. 90° Can you find the or area of \(180 \triangle R°''S? ''T4''. \) to What verify its congr happens ifuency you with perform a $$\triangle reflection RST first$$? ### and Tip then a: rotationAlways? 5. Can you write the transformation as apply a transformations step single composite rule by combining both rotation step to and reflection? **Tip avoid:** errors Always apply transformations step-by-step to. avoid errors.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Geometry
Rotation
Reflection
Coordinate Transformation
Formulas
90° counterclockwise rotation rule: (x, y) → (-y, x)
Reflection over the x-axis rule: (x, y) → (x, -y)
Theorems
Transformation of geometric shapes
Properties of rotations and reflections
Suitable Grade Level
Grades 7-9