Jerome claims he can find the area of the composite shape shown by inscribing it in a rectangle and subtracting. Kelly claims that to find the area you need to use addition. Who is correct? Justify your answer.The picture looks like a 2 sided pointed pencil with 6 sides. The length of the whole 2 sided pencil is 21 cm. The left pointed area is triangle with 4cm pointed to a dividing line. The sides are 13 cm.It is 6cm between each 13 cm sides.
I'd say either can be true, but not all shapes can be inscribed in a rectangle. That is, so that all their vertices touch the sides of the rectangle.
To determine who is correct between Jerome and Kelly, we need to understand how to find the area of the composite shape as described.
Jerome's approach suggests inscribing the shape in a rectangle and then subtracting to find the composite area. Kelly, on the other hand, claims that addition is needed to find the area.
To evaluate both methods, let's break down the composite shape into its individual components and calculate their areas.
1. Rectangle:
- Since the entire length of the 2-sided pencil is 21 cm, the width of the rectangle must be 21 cm to encompass the entire shape.
2. Left pointed area (triangle):
- The base of the triangle is the dividing line, which is 6 cm long.
- The height of the triangle is the distance between the dividing line and the apex of the left pointed area, which is 4 cm.
- Calculating the area of the triangle using the formula A = (base * height) / 2, we get: A = (6 cm * 4 cm) / 2 = 12 cm².
3. Middle section (rectangle):
- The width of the rectangle is 6 cm, which corresponds to the distance between each side of length 13 cm.
- The length of the rectangle is the remaining length of the 2-sided pencil after subtracting the base of the triangle on the left side. Therefore, it is 21 cm - 6 cm = 15 cm.
4. Right pointed area (triangle):
- The right pointed area is identical to the left pointed area in both base and height dimensions.
- Using the same calculations as before, the area of the triangle is: A = (6 cm * 4 cm) / 2 = 12 cm².
To find the composite area, we can use Jerome's method (subtracting) or Kelly's method (adding). Let's verify which approach leads to the correct answer.
Jerome's method:
- Calculate the total area of the individual components: Rectangle = 21 cm * 6 cm = 126 cm², Left Triangle = 12 cm², Middle Rectangle = 6 cm * 15 cm = 90 cm², Right Triangle = 12 cm².
- Subtract the areas of the triangles from the rectangle: Composite Area = Rectangle - Left Triangle - Right Triangle = 126 cm² - 12 cm² - 12 cm² = 102 cm².
Kelly's method:
- Calculate the total area of the individual components: Left Triangle + Middle Rectangle + Right Triangle = 12 cm² + 90 cm² + 12 cm² = 114 cm².
Based on the calculations, we find that Jerome's method of subtracting the areas of the triangles from the rectangle yields a composite area of 102 cm², while Kelly's method of adding the areas of the individual components gives a composite area of 114 cm².
Therefore, Jerome's claim is correct, and Kelly's claim is incorrect in this situation.