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 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 triangles apex, β at the square windows center, γ at the red hearts top vertex, and δ at the puppys head top, with all greek letter labels clearly placed next to their respective points.

Explanation:

Since this is a geometric construction problem, we'll use the Step - by - Step Format to outline the construction process.

Step 1: Draw the rectangular house body

Using the 1 cm grid paper and a ruler, draw a rectangle with a length of 8 cm (8 grid squares) and a width of 6 cm (6 grid squares). Outline this rectangle with a brown colored pencil/marker.

Step 2: Draw the isosceles triangle roof

The base of the isosceles triangle is the top side of the rectangular house body (8 cm long). The height of the triangle is 3 cm (3 grid squares). Find the mid - point of the top side of the rectangle. From this mid - point, draw a vertical line upwards for 3 cm. Then, connect the endpoints of the top side of the rectangle to the top of this vertical line to form the isosceles triangle. Fill this triangle completely with a light gray colored pencil/marker.

Step 3: Draw the square window

Locate the left - middle of the rectangular house body. A square with a side - length of 2 cm (2 grid squares) has its sides parallel to the sides of the rectangle. Draw this square and outline it with a blue colored pencil/marker.

Step 4: Draw the rectangular door

Locate the lower - middle of the rectangular house body. A rectangle with dimensions 2 cm (width, 2 grid squares) by 4 cm (height, 4 grid squares) has its sides parallel to the sides of the main rectangle. Draw this rectangle and outline it with a black colored pencil/marker.

Step 5: Construct the red heart

Inside the house, to the right of the window, construct a 4 cm (horizontal) × 4 cm (vertical) solid heart. You can construct a heart by first drawing two semi - circles on top of a downward - pointing triangle or by using a more geometric approximation (e.g., a combination of curves). Fill this heart completely with a red colored pencil/marker. Label the width/height of the heart as 4 cm × 4 cm.

Step 6: Draw the puppy

Adjacent to the red heart (lower position inside the house), draw a simple geometric puppy with a total height of at most 3 cm. You can start with a small rectangle or a circle for the body and a circle for the head. Outline the puppy with a black colored pencil/marker. Then, draw small round glasses on its face with a blue frame outline.

Step 7: Mark the Greek letters

• Mark \(\alpha\) at the apex of the roof triangle (the top - most point of the isosceles triangle).
• To find the center of the square window, find the mid - point of the square's diagonals (or the mid - point of its horizontal and vertical mid - lines). Mark \(\beta\) at this center.
• Find the top vertex of the red heart (the highest point of the heart's outline) and mark \(\gamma\) there.
• Find the top of the puppy's head (the highest point of the puppy's head outline) and mark \(\delta\) there. Make sure all Greek letter labels are clearly placed next to their respective points.

Answer:

The construction is carried out by following the above - mentioned steps on the 1 cm grid paper, using the appropriate colored pencils/markers and a ruler for precision, and marking the Greek letters at the specified points.