Question

ap geometry – constructed response question
total score: 10 points | time allowed: 15 minutes
materials: 1 cm grid paper, colored pencils/markers (brown, light gray, blue, red, black), ruler, pencil
directions: on 1 cm grid paper, perform a precise geometric construction that includes a house, solid heart, and simple puppy with glasses, following all color, fill, dimension, and labeling rules below; label all key linear dimensions directly on the figure, fill specified regions completely, and clearly mark the greek letters α, β, γ, δ at the required positions (partial credit for correct partial constructions). construct an 8 cm × 6 cm rectangular house body with a brown outline, topped with an isosceles triangle roof (3 cm height) that has the rectangle’s top side as its base and is fully filled with light gray; on the left middle of the rectangular body, draw a 2 cm side - length square window with a blue outline, and on the lower middle of the rectangular body, draw a 2 cm × 4 cm rectangular door with a black outline. inside the house, to the right of the window, construct a 4 cm (horizontal) × 4 cm (vertical) solid heart fully filled with red (label its width/height); adjacent to the red heart (lower position inside the house), draw a simple geometric puppy (total height ≤ 3 cm) with a black outline and add small round glasses on its face with a blue frame outline. mark α at the roof triangle’s apex, β at the square window’s center, γ at the red heart’s top vertex, and δ at the puppy’s head top, with all greek letter labels clearly placed next to their respective points.

Explanation:

Step1: Prepare Grid Paper

Use 1 cm grid paper. Each square is 1 cm × 1 cm.

Step2: Construct House Body

Draw an 8 cm (width) × 6 cm (height) rectangle (brown outline). The base is along the grid, height 6 cm (vertical), width 8 cm (horizontal).

Step3: Construct Roof (Isosceles Triangle)

The base of the triangle is the top side of the house (8 cm long). The height is 3 cm (vertical from base to apex). Draw the two equal sides from the base endpoints to the apex (light gray fill).

Step4: Draw Window (Square)

On the left middle of the house body: a square with side 2 cm (blue outline). Middle left: horizontal position ~2 - 4 cm from left (since house is 8 cm wide, middle left is around x=2 - 4, y=3 - 5 (house height 6 cm, middle vertical ~3 - 5 cm)).

Step5: Draw Door (Rectangle)

Lower middle of house body: 2 cm (width) × 4 cm (height) rectangle (black outline). Lower middle: y=0 - 4 cm (height 4 cm), x=3 - 5 cm (width 2 cm, house width 8 cm, middle x=3 - 5 cm).

Step6: Construct Solid Heart (Red Fill)

Inside the house, right of the window: 4 cm (horizontal) × 4 cm (vertical). The heart can be constructed by combining a square (4x4) with semicircular top and bottom, or using the grid to plot the heart shape (e.g., top and bottom curves). Label width 4 cm, height 4 cm (red fill).

Step7: Draw Puppy (Black Outline)

Adjacent to the red heart (lower position): total height ≤ 3 cm. Simple geometric shape (e.g., a circle for head, rectangle for body, triangles for ears; black outline). Add small round glasses (blue frame outline) on the face.

Step8: Mark Greek Letters

• α: At the roof triangle’s apex (top of the triangle).
• β: At the square window’s center (midpoint of the square: (x + 1, y + 1) if window is from (x, y) to (x+2, y+2)).
• γ: At the red heart’s top vertex (topmost point of the heart).
• δ: At the puppy’s head top (top of the puppy’s head, e.g., top of the circular head).

Step9: Fill and Label

• Fill roof (light gray), heart (red), window (blue outline, no fill), door (black outline, no fill), puppy (black outline, no fill), glasses (blue frame outline).
• Label heart’s width (4 cm) and height (4 cm) on the figure.

Answer:

The geometric construction is completed following the steps: house (8×6 rectangle, brown), roof (isosceles triangle, 8 cm base, 3 cm height, light gray), window (2×2 square, blue), door (2×4 rectangle, black), heart (4×4, red, labeled 4 cm×4 cm), puppy (≤3 cm height, black, blue glasses), and Greek letters at specified positions.