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 rectangles 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 windows 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.

Answer:

Step-by-Step Construction Guide:

Step 1: Prepare Grid Paper

Use 1 cm grid paper. Each square on the grid represents 1 cm.

Step 2: Draw the House Body

• Rectangle Dimensions: 8 cm (width) × 6 cm (height).
• Outline Color: Brown.
• Locate the rectangle: Start at a grid point (e.g., (0,0)) and draw to (8,6) (using grid lines for precision).

Step 3: Draw the Roof (Isosceles Triangle)

• Base: The top side of the house body (length = 8 cm).
• Height: 3 cm (from the base to the apex).
• Fill Color: Light gray.
• Find the midpoint of the house’s top side (at 4 cm from the left/right). From this midpoint, draw a vertical line 3 cm up (to (4, 6+3) = (4,9)). Connect this apex to the two top corners of the house body ( (0,6) and (8,6) ) to form the isosceles triangle.

Step 4: Draw the Window

• Shape: Square with side length 2 cm.
• Outline Color: Blue.
• Position: Left middle of the house body. The house body is 6 cm tall, so the middle is at 3 cm height. The window’s top-left corner: (0, 6 - 3 - 1) = (0,2)? Wait, better: House body is from y=0 to y=6 (bottom to top). Middle vertically: 3 cm from bottom (y=3) to 3 cm from top (y=3). Wait, house body height is 6 cm, so middle is at y=3 (since 0 to 6, midpoint at 3). The window is a 2x2 square, so its center is at (x, 3), where x is left middle. Let’s place the window 1 cm from the left edge of the house (so left edge of window at x=1, right at x=3; bottom at y=2, top at y=4). Thus, center of window is at (2, 3) (since (1+3)/2 = 2, (2+4)/2 = 3).

Step 5: Draw the Door

• Shape: Rectangle with dimensions 2 cm (width) × 4 cm (height).
• Outline Color: Black.
• Position: Lower middle of the house body. House body height is 6 cm, so “lower middle” is near the bottom. Let’s center it horizontally: house width is 8 cm, so door width 2 cm → centered at x=4 (from x=3 to x=5). Vertically: bottom at y=0, top at y=4 (since 4 cm height). So door spans (3,0) to (5,4).

Step 6: Construct the Solid Heart

• Dimensions: 4 cm (horizontal) × 4 cm (vertical).
• Fill Color: Red.
• Position: Inside the house, to the right of the window.
• A “solid heart” can be constructed as two semicircles on top of a square (or a symmetric shape). For simplicity:
• Draw a 4x4 square (from x=4, y=2 to x=8, y=6? Wait, no: inside the house (x=0 to 8, y=0 to 6). To the right of the window (window is at x=0-3, y=2-4). So heart is at x=4-8, y=2-6? Wait, heart dimensions: 4 cm horizontal (width) and 4 cm vertical (height). Let’s set the heart’s center at (6, 4) (right of window’s center (2,3)). The heart’s width is 4 cm (so from x=4 to x=8) and height 4 cm (from y=2 to y=6). Draw the heart: top as two semicircles (left and right) with radius 2 cm, bottom as a square or straight line? Alternatively, use the grid to draw a heart shape: two upper semicircles (left and right) and a lower triangle or square. Ensure it’s 4x4 and filled red. Label width (4 cm) and height (4 cm) on the figure.

Step 7: Draw the Puppy

• Total Height: ≤ 3 cm.
• Outline Color: Black.
• Position: Adjacent to the red heart (lower position inside the house). So below the heart (heart is from y=2 to y=6, so puppy is from y=0 to y=3).
• Shape: Simple geometric (e.g., a circle for the head, a rectangle for the body, triangles for ears).
• Glasses: Small round (semicircles or circles) with blue frame outline, on the face.

Step 8: Label Greek Letters

• α: At the roof triangle’s apex (top of the isosceles triangle, at (4,9)).
• β: At the square window’s center (at (2, 3)).
• γ: At the red heart’s top vertex (top of the heart, e.g., at (6, 6) if heart is 4x4 from y=2 to y=6).
• δ: At the puppy’s head top (e.g., if puppy’s head is a circle at (7, 3), top at (7, 3 + radius); adjust based on puppy’s design).

Step 9: Fill and Label

• Fill the roof with light gray, the heart with red.
• Label all key linear dimensions (house: 8 cm × 6 cm, roof height: 3 cm, window: 2 cm, door: 2 cm × 4 cm, heart: 4 cm × 4 cm, puppy height: ≤3 cm) directly on the figure.
• Ensure all outlines (brown, blue, black, blue for glasses) are clear, and Greek letters are marked next to their points.

Final Check:

• Verify all shapes: house (8x6), roof (8x3 triangle), window (2x2 square), door (2x4 rectangle), heart (4x4), puppy (≤3 cm tall).
• Check colors: brown (house), light gray (roof), blue (window, glasses), black (door, puppy), red (heart).
• Confirm positions: window left middle, door lower middle, heart right of window, puppy below heart.
• Mark α, β, γ, δ at correct points.

(Note: This is a construction guide; actual drawing requires using the grid paper and tools as specified.)