Question

select the correct answer from each dropdown menu.
getting ready for a big car race, kenna decided to make five racing flags to put at each turn of the racetrack. kenna sketched the flag design in a coordinate plane to help determine how much material she will need for the flag.
assuming each grid mark represents a foot, how much material will kenna need to make each flag and all five flags?
kenna will need square feet of material for one flag.
thus, kenna will need square feet of material to make all five flags.

Explanation:

Step1: Determine the base and height of the triangle

Looking at the coordinate plane, the vertical segment (height) from \( y = 4 \) to \( y = 7 \) (assuming the top vertex is at \( (6,7) \), the bottom vertex at \( (5,4) \), and the other vertex at \( (7,5) \)? Wait, actually, let's check the grid. The base: from \( x = 5 \) to \( x = 7 \), so base length is \( 7 - 5 = 2 \)? Wait, no, maybe the triangle has a base along the vertical? Wait, the vertical line is from \( (5,4) \) to \( (5,7) \)? Wait, no, the figure is a triangle. Let's find the coordinates. Let's assume the vertices are \( (5,4) \), \( (6,7) \), and \( (7,5) \)? Wait, maybe easier: the base of the triangle (horizontal or vertical). Wait, each grid mark is 1 foot. Let's see the triangle: the vertical side from \( (5,4) \) to \( (5,7) \) is length \( 7 - 4 = 3 \) feet? No, wait, the height: maybe the base is 2 units (from \( x = 5 \) to \( x = 7 \), so 2 feet) and height is 3 units (from \( y = 4 \) to \( y = 7 \), 3 feet)? Wait, no, area of a triangle is \( \frac{1}{2} \times base \times height \). Let's look at the grid. The triangle: let's find the base and height. Let's take the base as the horizontal distance between two points. Wait, the left vertex is at \( (5,4) \), the right vertex at \( (7,5) \)? No, maybe the triangle has a base of 2 (from \( x = 5 \) to \( x = 7 \), so 2 feet) and height of 3 (from \( y = 4 \) to \( y = 7 \), 3 feet). Wait, no, let's count the grid. The vertical segment from \( (5,4) \) to \( (5,7) \) is 3 units (since 7 - 4 = 3). The horizontal segment from \( (5,4) \) to \( (7,4) \) would be 2 units, but the triangle is above. Wait, actually, the triangle is a triangle with base length 2 (from \( x = 5 \) to \( x = 7 \)) and height 3 (from \( y = 4 \) to \( y = 7 \)). So area of one triangle is \( \frac{1}{2} \times base \times height = \frac{1}{2} \times 2 \times 3 = 3 \)? No, that can't be. Wait, maybe the base is 2 and height is 3? Wait, no, let's check the coordinates again. Let's assume the three vertices are \( (5,4) \), \( (6,7) \), and \( (7,4) \). Then the base is from \( (5,4) \) to \( (7,4) \), which is 2 units (length 2), and the height is from \( y = 4 \) to \( y = 7 \), which is 3 units. Then area is \( \frac{1}{2} \times 2 \times 3 = 3 \)? No, that seems too small. Wait, maybe the base is 2 and height is 3? Wait, no, maybe the vertical side is 3 and horizontal side is 2. Wait, maybe I made a mistake. Wait, the grid: each grid mark is 1 foot. Let's look at the triangle: the left side is at \( x = 5 \), from \( y = 4 \) to \( y = 7 \) (length 3). The right side has a vertex at \( (7,5) \)? Wait, no, the triangle is a triangle with base 2 (from \( x = 5 \) to \( x = 7 \)) and height 3 (from \( y = 4 \) to \( y = 7 \)). Wait, no, area of a triangle is \( \frac{1}{2} \times base \times height \). Wait, maybe the base is 2 and height is 3, so area is \( \frac{1}{2} \times 2 \times 3 = 3 \)? No, that doesn't seem right. Wait, maybe the base is 2 and height is 3, but that would be 3. Wait, no, maybe the base is 2 and height is 3, so 3 square feet per flag? Then five flags would be 15. But that seems too small. Wait, maybe I messed up the coordinates. Let's re-examine the graph. The triangle: the bottom vertex is at \( (5,4) \), the top vertex at \( (6,7) \), and the right vertex at \( (7,5) \). Let's use the formula for the area of a triangle with coordinates. The coordinates are \( (5,4) \), \( (6,7) \), \( (7,5) \). Using the shoelace formula: \( \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)| \). Plugging in: \( \frac{1}{2} |5(7 - 5) + 6(5 - 4) + 7(4 - 7)| = \frac{1}{2} |5(2) + 6(1) + 7(-3)| = \frac{1}{2} |10 + 6 - 21| = \frac{1}{2} |-5| = \frac{5}{2} = 2.5 \)? No, that's not right. Wait, maybe the triangle is a right triangle. Let's see: from \( (5,4) \) to \( (6,7) \) is a vertical or horizontal? No, \( (5,4) \) to \( (7,4) \) is horizontal (length 2), and \( (7,4) \) to \( (6,7) \) is vertical? No, \( (7,4) \) to \( (6,7) \) has a vertical change of 3 and horizontal change of -1. Wait, maybe the base is 2 (from \( (5,4) \) to \( (7,4) \)) and height is 3 (from \( (7,4) \) to \( (6,7) \) vertical distance? No, the height would be the vertical distance from the top vertex to the base. The base is \( y = 4 \), from \( x = 5 \) to \( x = 7 \). The top vertex is at \( (6,7) \), so the vertical distance from \( y = 4 \) to \( y = 7 \) is 3. So the base is 2 (length from \( x = 5 \) to \( x = 7 \)) and height is 3 (vertical distance from base to top vertex). So area is \( \frac{1}{2} \times 2 \times 3 = 3 \) square feet per flag. Then five flags would be \( 3 \times 5 = 15 \) square feet. Wait, but maybe I made a mistake. Alternatively, maybe the base is 2 and height is 3, so 3 per flag, 15 for five. Or maybe the base is 2 and height is 3, so 3. Wait, let's check again. Each grid mark is 1 foot. The triangle: the horizontal length from \( x = 5 \) to \( x = 7 \) is 2 feet (base). The vertical length from \( y = 4 \) to \( y = 7 \) is 3 feet (height). So area of triangle is \( \frac{1}{2} \times base \times height = \frac{1}{2} \times 2 \times 3 = 3 \) square feet for one flag. Then for five flags, \( 3 \times 5 = 15 \) square feet. Wait, but maybe the base is 2 and height is 3, so that's correct. Alternatively, maybe the base is 2 and height is 3, so 3. So the first dropdown is 3, the second is 15. Wait, but let's confirm. If the triangle has base 2 and height 3, area is 3. So one flag: 3 square feet, five flags: 15 square feet.

Step2: Calculate material for five flags

Multiply the area of one flag by 5: \( 3 \times 5 = 15 \).

Answer:

For one flag: 3 square feet. For five flags: 15 square feet. So the first dropdown is 3, the second is 15.