Enlarging a Pennant Mural on the Coordinate Plane
The Riverside Middle School art club is painting a triangular pennant mural in a hallway. They draft the design on a coordinate plane where each grid unit represents 4 inches, with vertices A(2, 1), B(8, 1), and C(2, 7). The club votes to enlarge the pennant by a scale factor of 2.5 using point A as the center of dilation. A parent volunteer argues the enlarged pennant is congruent to the draft because “the angles never change.” Meanwhile, a second group wants to reposition a copy of the original pennant by rotating it 90° counterclockwise about the origin and then reflecting that image across the y-axis; they claim this composition is the same as a single reflection. The club has 6 feet of wall height available and must confirm the enlarged pennant fits there before anyone buys paint.
1. Find the vertices of the dilated pennant, compare side lengths and areas of the draft and the image, and rule on the parent's congruence claim. Will the enlarged pennant fit within the 6-foot wall height?
Read the worked answer
Dilating from center A(2, 1) means multiplying each vector from A by 2.5.
- A stays at (2, 1).
- Vector A→B is (6, 0); times 2.5 gives (15, 0), so B′ = (17, 1).
- Vector A→C is (0, 6); times 2.5 gives (0, 15), so C′ = (2, 16).
Side checks: AB = 6 units and A′B′ = 15 units; AC = 6 and A′C′ = 15. Every side is multiplied by 2.5 and every angle is preserved, which is the definition of similarity - not congruence. The parent is right that angles are unchanged but wrong that the figures are congruent, because congruent figures must have equal side lengths.
Area: draft = ½(6)(6) = 18 square units; image = ½(15)(15) = 112.5 square units, which is 6.25 = 2.5² times larger. Area scales by the square of the factor, not by the factor itself.
Height: the image rises 16 − 1 = 15 units, and 15 × 4 inches = 60 inches = 5 feet, so it fits under the 6-foot ceiling with about 1 foot to spare.
2. A second group claims that rotating the draft 90° counterclockwise about the origin and then reflecting across the y-axis is equivalent to one reflection. Test the claim with a specific vertex, and state either the rule and line of reflection or the refutation.
Read the worked answer
Follow the rules one step at a time. A 90° counterclockwise rotation about the origin sends (x, y) to (−y, x). Reflecting across the y-axis sends (x, y) to (−x, y). Composing them: (x, y) → (−y, x) → (y, x).
That final rule, (x, y) → (y, x), is exactly reflection across the line y = x.
Test with B(8, 1): rotation gives (−1, 8); reflecting across the y-axis gives (1, 8). Reflecting B directly across y = x also gives (1, 8). Test with C(2, 7): rotation gives (−7, 2); the y-axis reflection gives (7, 2); reflection across y = x gives (7, 2) as well.
So the claim is correct that the composition is a single reflection, but the line is y = x, not the y-axis. Only points already on y = x stay fixed. A useful check: the composition swaps coordinates twice-over in a way that reverses orientation, which is consistent with a reflection rather than a rotation.