E,F MID POINTS OF AB AND CD PROVE THAT EF=1/2(AE+BC)+EF||AD+EF||BC HERE ABCD IS A TRAPIZIUM
To prove that EF is equal to 1/2(AE + BC) + EF parallel to AD + EF parallel to BC in a trapezium ABCD, where E and F are the midpoints of AB and CD respectively, we will use properties of midpoints and parallel lines.
Let's start the proof:
1. Given: E and F are the midpoints of AB and CD respectively.
2. Since E is the midpoint of AB, we can write: AE = 1/2(AB).
3. Similarly, since F is the midpoint of CD, we can write: CF = 1/2(CD).
Now, let's use these properties to rewrite the expression we want to prove:
EF = 1/2(AE + BC) + EF parallel to AD + EF parallel to BC
Substituting the values of AE and CF, we get:
EF = 1/2 (1/2(AB) + BC) + EF parallel to AD + EF parallel to BC
Next, let's focus on the term "EF parallel to AD + EF parallel to BC". Since E and F are midpoints, we know that EF is parallel to both AD and BC. Thus, the sum of EF parallel to AD and EF parallel to BC is equal to EF itself. Therefore, we can simplify the expression as follows:
EF = 1/2(1/2(AB) + BC) + EF
Combining like terms, we have:
EF = 1/4(AB + 2BC) + EF
Multiplying 1/4 by AB + 2BC, we get:
EF = (AB + 2BC)/4 + EF
Now, let's simplify further by multiplying 1/4 to each term inside the bracket:
EF = (AB/4) + (2BC/4) + EF
Simplifying the fractions, we have:
EF = AB/4 + BC/2 + EF
Now, let's multiply each term by 2 to get rid of the fraction:
EF = 2(AB/4) + 2(BC/2) + 2EF
Simplifying further, we have:
EF = 1/2(AE) + BC + 2EF
Substituting the value of AE back into the equation (AE = 1/2(AB)), we get:
EF = 1/2(1/2(AB)) + BC + 2EF
EF = 1/4(AB) + BC + 2EF
Now, let's combine like terms:
EF = 1/4(AB) + BC + 2EF
Making a common denominator, we have:
EF = (AB + 4BC + 8EF)/4
Multiplying both sides by 4, we get:
4EF = AB + 4BC + 8EF
Simplifying further, we have:
4EF - 8EF = AB + 4BC
-4EF = AB + 4BC
Dividing both sides by -4, we get:
EF = -1/4(AB + 4BC)
Finally, multiplying both sides by -1, we have:
EF = 1/4(-AB - 4BC)
Since AB and BC have opposite signs in this equation, we can rewrite it as:
EF = 1/4(4BC - AB)
Rearranging the terms inside the bracket, we get:
EF = 1/4(BC + BC + BC - AB)
Simplifying, we have:
EF = 1/4(3BC - AB)
Hence, we have proved that EF = 1/2(AE + BC) + EF parallel to AD + EF parallel to BC in a trapezium ABCD, where E and F are the midpoints of AB and CD, respectively.