Question

solve for x. the polygons in each pair are sin
24)
10 15
x+4 21
10 15
write a rule to describe each transformation.
25)
a) rotation 180° about the origin
b) reflection across the x - axis
c) rotation 90° clockwise about the origin
d) translation: 1 unit right and 2 units down

Explanation:

Problem 24: Solve for \( x \) (Similar Polygons)

Step 1: Identify Proportions

Since the polygons are similar, corresponding sides are proportional. The sides of length 10 correspond to 15, and \( x + 4 \) corresponds to 21. So, \(\frac{10}{15}=\frac{x + 4}{21}\).

Step 2: Simplify and Solve

Simplify \(\frac{10}{15}\) to \(\frac{2}{3}\). Then, \(\frac{2}{3}=\frac{x + 4}{21}\). Cross - multiply: \(2\times21 = 3\times(x + 4)\). So, \(42=3x + 12\). Subtract 12 from both sides: \(42-12 = 3x\), \(30 = 3x\). Divide by 3: \(x = 10\).

• Option A: A \(180^{\circ}\) rotation about the origin changes \((x,y)\) to \((-x,-y)\). Checking the coordinates of the vertices (e.g., \(U\) and \(U'\), \(T\) and \(T'\), \(V\) and \(V'\)) shows this matches.
• Option B: Reflection over \(x\) - axis changes \((x,y)\) to \((x,-y)\), which doesn't match the transformation.
• Option C: \(90^{\circ}\) clockwise rotation about the origin changes \((x,y)\) to \((y,-x)\), which doesn't match.
• Option D: Translation would move all points by the same vector, but the transformation here is rotational, not translational.

Answer:

\(x = 10\)

Problem 25: Transformation Rule