Use the image to answer the question.
An illustration shows a composite structure of a pentagon with two right angles and a square, resembling a house. The base of the pentagon is of length 6. The two sides are perpendicular to the base. The perpendicular right side of the pentagon shares a common side with a square of side 3, with the base of the square continuing out length 3 from the end of the pentagon base of length 6. A solid vertical line is drawn from the top vertex of the pentagon to the midpoint on the base, dividing the shape in half. The height from the top vertex to the end of the perpendicular sides is marked as 3.
A home remodeler has chosen a tile pattern for a floor. How much tile is needed to create the pattern?
(1 point)
square centimeters
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page 15 of 16
To determine the total area of the floor that needs to be tiled, we need to calculate the area of the pentagon and the square separately and then add them together.
Area of the pentagon:
First, we need to find the area of the triangle formed by the top vertex, the midpoint on the base, and the end of the perpendicular sides. This triangle is a right triangle with a base of 3 and a height of 3. So, the area of the triangle is (1/2) * base * height = (1/2) * 3 * 3 = 4.5 square units.
Next, we need to find the area of the trapezoid formed by the base of the pentagon, the base of the square, and the two perpendicular sides. The trapezoid can be divided into a rectangle with dimensions 6 (base of pentagon) and 3 (height) and a right triangle with base 3 and height 3. The area of the rectangle is 6 * 3 = 18 square units, and the area of the right triangle is (1/2) * 3 * 3 = 4.5 square units. So, the total area of the trapezoid is 18 + 4.5 = 22.5 square units.
Adding the area of the triangle and the trapezoid, the total area of the pentagon is 4.5 + 22.5 = 27 square units.
Area of the square:
The square has a side length of 3, so its area is 3 * 3 = 9 square units.
Total tile needed for the pattern:
Adding the area of the pentagon and the square, the total tile needed to create the pattern is 27 + 9 = 36 square units.
Therefore, the remodeler would need 36 square centimeters of tile to create the pattern on the floor.