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:

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

Step 1: Draw the house body

Draw a rectangle with length \( 8 \, \text{cm} \) and width \( 6 \, \text{cm} \) using a ruler on the grid paper. Outline it with brown.

Step 2: Draw the roof

The roof is an isosceles triangle with the base equal to the top side of the rectangle ( \( 8 \, \text{cm} \)) and height \( 3 \, \text{cm} \). Draw the triangle on top of the rectangle and fill it with light gray. Mark \( \alpha \) at the apex 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 (midpoint of the diagonals or midpoint of the sides) and mark \( \beta \) there.

Step 4: Draw the rectangular door

On the lower middle of the rectangular body, draw a rectangle with length \( 4 \, \text{cm} \) and width \( 2 \, \text{cm} \). Outline it with black.

Step 5: Construct the red heart

Inside the house, to the right of the window, construct a \( 2 \, \text{cm} \) (horizontal) \( \times 4 \, \text{cm} \) (vertical) solid heart and fill it with red. Find the top vertex of the heart and mark \( \gamma \) there.

Step 6: Draw the puppy

Adjacent to the red heart (lower position inside the house), draw a simple geometric puppy with total height \( \leq 3 \, \text{cm} \) with a black outline. Add small round glasses on its face with a blue frame outline. Find the top of the puppy's head and mark \( \delta \) there.

(Note: Since this is a construction problem, the final answer is the completed geometric figure with all the required elements drawn, labeled, and colored as per the instructions. The above steps guide you to create the figure.)

Answer:

The completed geometric construction as per the given instructions (with the house, roof, window, door, heart, puppy, and Greek letter labels in their respective positions).