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 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:

Since this is a geometric construction problem, we can break down the steps to perform the construction:

Step 1: Draw the rectangular house body

Using the grid paper, draw a rectangle with length \( 8 \, \text{cm} \) and width \( 6 \, \text{cm} \). Outline it with brown.

Step 2: Construct the isosceles triangle roof

The base of the isosceles triangle is the top side of the rectangle (length \( 8 \, \text{cm} \)). The height of the triangle is \( 3 \, \text{cm} \). Find the midpoint of the top side of the rectangle, then draw two line segments from this midpoint to the endpoints of the top side, forming the isosceles triangle. Fill the triangle with light gray. Mark \( \alpha \) at the apex (top vertex) of the triangle.

Step 3: Draw the square window

On the left middle of the rectangular body, draw a square with side - length \( 2 \, \text{cm} \). Outline it with blue. Find the center of the square (the intersection of its diagonals) and mark \( \beta \) there.

Step 4: Draw the rectangular door

On the lower middle of the rectangular body, draw a rectangle with dimensions \( 2 \, \text{cm}\times4 \, \text{cm} \). Outline it with black.

Step 5: Construct the red heart

To the right of the window, inside the house, construct a \( 4 \, \text{cm} \) (horizontal) \( \times4 \, \text{cm} \) (vertical) solid heart. Fill it with red. Find the top vertex of the heart and mark \( \gamma \) there.

Step 6: Draw the puppy with glasses

Adjacent to the red heart (lower position inside the house), draw a simple geometric puppy with a total height \( \leq3 \, \text{cm} \). Outline the puppy with black. Draw small round glasses on its face with a blue frame outline. Find the top of the puppy's head and mark \( \delta \) there.

Answer:

To solve this geometric construction problem, follow the step - by - step instructions above to draw the house, roof, window, door, heart, puppy, and mark the Greek letters at the specified positions on the \( 1 \, \text{cm} \) grid paper using the appropriate colors and outlines.