The area of rectangle ACED is 80 square centimeters, and the area of triangle BCE is 30 square centimeters, If AD = 4 centimeters, then what is AB ?
a. 5 cm
b. 10 cm
c. 15 cm
d. 20 cm
e. 25 cm
please answer and explain
AC*AD = 80, so AC = 20
AD=CE=4
CE*BC/2 = 30, so BC=15
AB+BC=20, so AB=5
To find the length of AB, we need to carefully analyze the given information and utilize the formulas for calculating the area of a rectangle and a triangle. Let's break down the problem step by step.
1. Area of Rectangle ACED: The area of a rectangle is calculated by multiplying its length (AD) by its width (CE). We are given that the area of rectangle ACED is 80 square centimeters. Therefore, we can write the equation: AD * CE = 80.
2. Area of Triangle BCE: The area of a triangle is calculated by multiplying the base (CE) by the height (BE) and then dividing the result by 2. We are given that the area of triangle BCE is 30 square centimeters. Therefore, we can write the equation: (CE * BE) / 2 = 30.
3. We can solve for CE using equation (2): CE * BE = 60. To simplify things, let's solve equation (2) for CE: CE = (2 * 30) / BE = 60 / BE.
4. Substitute the value of CE in equation (1): AD * (60 / BE) = 80.
5. We are given that AD = 4 centimeters. We can substitute this value into equation (4): 4 * (60 / BE) = 80.
6. Simplify equation (5): (240 / BE) = 80.
7. Multiply both sides of equation (6) by BE: 240 = 80 * BE.
8. Divide both sides of equation (7) by 80: BE = 240 / 80 = 3 centimeters.
9. Now we know that BE = 3 centimeters. To find AB, we need to subtract AC from AD. Since AC = BE, AB = AD - AC = 4 - 3 = 1 centimeter.
Therefore, the length of AB is 1 centimeter, which is not one of the given answer options. It's possible that there may be an error or inconsistency in the information provided.