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'll break down the steps to perform the construction:

Step 1: Draw the Rectangular House Body

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

Step 2: Construct the Isosceles Triangle Roof

The base of the isosceles triangle is the top side of the rectangular house (length \( 8 \, \text{cm} \)). The height of the triangle is \( 3 \, \text{cm} \) (3 grid squares). Find the midpoint of the top side of the rectangle (at \( 4 \, \text{cm} \) from either end) and draw a point \( 3 \, \text{cm} \) above this midpoint. Connect this point to the two endpoints of the top side of the rectangle to form the isosceles triangle. Fill this triangle with light gray.

Step 3: Draw the Square Window

On the left middle of the rectangular body, draw a square with side length \( 2 \, \text{cm} \) (2 grid squares). The "left middle" means the square is centered vertically on the left half of the rectangle. Outline this square with blue.

Step 4: Draw the Rectangular Door

On the lower middle of the rectangular body, draw a rectangle with length \( 2 \, \text{cm} \) and height \( 4 \, \text{cm} \) (2 grid squares by 4 grid squares). Outline this rectangle with black.

Step 5: Construct the Red Heart

Inside the house, to the right of the window, construct a \( 4 \, \text{cm} \) (horizontal) \( \times 4 \, \text{cm} \) (vertical) solid heart. You can construct a heart by drawing two semicircles on top of a downward - pointing triangle or using a standard heart - shape construction (e.g., two overlapping circles and a triangle at the bottom). Fill this heart with red and label its width/height.

Step 6: Draw the Puppy and Glasses

Adjacent to the red heart (lower position inside the house), draw a simple geometric puppy with a total height \( \leq 3 \, \text{cm} \). A simple puppy can be constructed using basic shapes: a circle or oval for the head (height \( \leq 3 \, \text{cm} \)) and a rectangle or triangle for the body. Outline the puppy with black. For the glasses, draw two small circles (or ovals) for the lenses and a blue - outlined frame connecting them on the puppy's face.

Step 7: Mark the Greek Letters

• \( \alpha \): Place this label next to the apex (top point) of the roof triangle.
• \( \beta \): Find the center of the square window (the intersection of its diagonals) and place \( \beta \) next to this center.
• \( \gamma \): Place \( \gamma \) next to the top vertex of the red heart.
• \( \delta \): Place \( \delta \) next to the top of the puppy's head.

(Note: Since this is a construction problem, the final "answer" is the completed geometric construction as per the above steps. If you were to submit this, you would present the grid - paper drawing with all the elements as described.)

Answer:

Step 1: Draw the Rectangular House Body

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

Step 2: Construct the Isosceles Triangle Roof

The base of the isosceles triangle is the top side of the rectangular house (length \( 8 \, \text{cm} \)). The height of the triangle is \( 3 \, \text{cm} \) (3 grid squares). Find the midpoint of the top side of the rectangle (at \( 4 \, \text{cm} \) from either end) and draw a point \( 3 \, \text{cm} \) above this midpoint. Connect this point to the two endpoints of the top side of the rectangle to form the isosceles triangle. Fill this triangle with light gray.

Step 3: Draw the Square Window

On the left middle of the rectangular body, draw a square with side length \( 2 \, \text{cm} \) (2 grid squares). The "left middle" means the square is centered vertically on the left half of the rectangle. Outline this square with blue.

Step 4: Draw the Rectangular Door

On the lower middle of the rectangular body, draw a rectangle with length \( 2 \, \text{cm} \) and height \( 4 \, \text{cm} \) (2 grid squares by 4 grid squares). Outline this rectangle with black.

Step 5: Construct the Red Heart

Inside the house, to the right of the window, construct a \( 4 \, \text{cm} \) (horizontal) \( \times 4 \, \text{cm} \) (vertical) solid heart. You can construct a heart by drawing two semicircles on top of a downward - pointing triangle or using a standard heart - shape construction (e.g., two overlapping circles and a triangle at the bottom). Fill this heart with red and label its width/height.

Step 6: Draw the Puppy and Glasses

Adjacent to the red heart (lower position inside the house), draw a simple geometric puppy with a total height \( \leq 3 \, \text{cm} \). A simple puppy can be constructed using basic shapes: a circle or oval for the head (height \( \leq 3 \, \text{cm} \)) and a rectangle or triangle for the body. Outline the puppy with black. For the glasses, draw two small circles (or ovals) for the lenses and a blue - outlined frame connecting them on the puppy's face.

Step 7: Mark the Greek Letters

• \( \alpha \): Place this label next to the apex (top point) of the roof triangle.
• \( \beta \): Find the center of the square window (the intersection of its diagonals) and place \( \beta \) next to this center.
• \( \gamma \): Place \( \gamma \) next to the top vertex of the red heart.
• \( \delta \): Place \( \delta \) next to the top of the puppy's head.

(Note: Since this is a construction problem, the final "answer" is the completed geometric construction as per the above steps. If you were to submit this, you would present the grid - paper drawing with all the elements as described.)