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 use the Step - by - Step Format to outline the construction process:

Step 1: Draw the rectangular house body

Using a ruler, draw a rectangle with length \(8\space cm\) and width \(6\space cm\). Outline this rectangle with a brown colored pencil or marker.

Step 2: Construct the isosceles triangle roof

The base of the isosceles triangle is the top side of the rectangular house (length \(8\space cm\)). The height of the triangle is \(3\space cm\). Find the mid - point of the top side of the rectangle. From this mid - point, draw a vertical line segment of length \(3\space cm\) upwards. Then connect the endpoints of this vertical segment to the endpoints of the top side of the rectangle to form the isosceles triangle. Fill the triangle with light gray. Mark the apex of this triangle (the top of the \(3\space cm\) vertical segment) with \(\alpha\).

Step 3: Draw the square window

Locate the left - middle of the rectangular house body. Draw a square with side - length \(2\space cm\). Outline this square with a blue colored pencil or marker. To find the center of the square, draw the diagonals of the square. The intersection point of the diagonals is the center. Mark this center with \(\beta\).

Step 4: Draw the rectangular door

Find the lower - middle of the rectangular house body. Draw a rectangle with length \(4\space cm\) and width \(2\space cm\). Outline this rectangle with a black colored pencil or marker.

Step 5: Construct the red heart

Inside the house, to the right of the window, construct a \(2\space cm\times4\space cm\) (horizontal \(\times\) vertical) red heart. First, we can think of the heart as a combination of two semi - circles and a rectangle (or use a more standard heart - shape construction). Fill the heart with red. Label the width and height of the heart as \(2\space cm\) (width) and \(4\space cm\) (height) respectively. Then, find the top vertex of the red heart and mark it with \(\gamma\).

Step 6: Draw the geometric puppy

Adjacent to the red heart (lower position inside the house), draw a simple geometric puppy with a total height \(\leq3\space cm\). Outline the puppy with a black colored pencil or marker. The puppy can be constructed using basic geometric shapes (e.g., a circle for the head, rectangles for the body and legs, triangles for the ears).

Step 7: Add the glasses to the puppy

Draw small round glasses on the puppy's face. The frame of the glasses should have a blue outline.

Step 8: Construct the solid heart

To the right of the window (inside the house), construct a \(4\space cm\) (horizontal) \(\times4\space cm\) (vertical) solid heart. Fill this heart with red.

Step 9: Mark the Greek letters for the heart and the puppy

Find the top vertex of the red heart and mark it with \(\gamma\). Find the top of the puppy's head and mark it with \(\delta\). Also, add small round glasses on the puppy's face with a blue frame outline.

Please note that this is a construction - based problem, and the above steps are a general guide. The actual construction may require more precise measurement and drawing techniques using the grid - paper and drawing tools as specified. If you are asked to actually perform the construction, you need to use the appropriate drawing instruments (ruler, colored pencils/markers) on the \(1\space cm\) grid paper.

Answer:

Step 1: Draw the rectangular house body

Using a ruler, draw a rectangle with length \(8\space cm\) and width \(6\space cm\). Outline this rectangle with a brown colored pencil or marker.

Step 2: Construct the isosceles triangle roof

The base of the isosceles triangle is the top side of the rectangular house (length \(8\space cm\)). The height of the triangle is \(3\space cm\). Find the mid - point of the top side of the rectangle. From this mid - point, draw a vertical line segment of length \(3\space cm\) upwards. Then connect the endpoints of this vertical segment to the endpoints of the top side of the rectangle to form the isosceles triangle. Fill the triangle with light gray. Mark the apex of this triangle (the top of the \(3\space cm\) vertical segment) with \(\alpha\).

Step 3: Draw the square window

Locate the left - middle of the rectangular house body. Draw a square with side - length \(2\space cm\). Outline this square with a blue colored pencil or marker. To find the center of the square, draw the diagonals of the square. The intersection point of the diagonals is the center. Mark this center with \(\beta\).

Step 4: Draw the rectangular door

Find the lower - middle of the rectangular house body. Draw a rectangle with length \(4\space cm\) and width \(2\space cm\). Outline this rectangle with a black colored pencil or marker.

Step 5: Construct the red heart

Inside the house, to the right of the window, construct a \(2\space cm\times4\space cm\) (horizontal \(\times\) vertical) red heart. First, we can think of the heart as a combination of two semi - circles and a rectangle (or use a more standard heart - shape construction). Fill the heart with red. Label the width and height of the heart as \(2\space cm\) (width) and \(4\space cm\) (height) respectively. Then, find the top vertex of the red heart and mark it with \(\gamma\).

Step 6: Draw the geometric puppy

Adjacent to the red heart (lower position inside the house), draw a simple geometric puppy with a total height \(\leq3\space cm\). Outline the puppy with a black colored pencil or marker. The puppy can be constructed using basic geometric shapes (e.g., a circle for the head, rectangles for the body and legs, triangles for the ears).

Step 7: Add the glasses to the puppy

Draw small round glasses on the puppy's face. The frame of the glasses should have a blue outline.

Step 8: Construct the solid heart

To the right of the window (inside the house), construct a \(4\space cm\) (horizontal) \(\times4\space cm\) (vertical) solid heart. Fill this heart with red.

Step 9: Mark the Greek letters for the heart and the puppy

Find the top vertex of the red heart and mark it with \(\gamma\). Find the top of the puppy's head and mark it with \(\delta\). Also, add small round glasses on the puppy's face with a blue frame outline.

Please note that this is a construction - based problem, and the above steps are a general guide. The actual construction may require more precise measurement and drawing techniques using the grid - paper and drawing tools as specified. If you are asked to actually perform the construction, you need to use the appropriate drawing instruments (ruler, colored pencils/markers) on the \(1\space cm\) grid paper.