Question

OAN, OMB, APB and MPN are straight lines and AN = 20A.

M is the midpoint of OB.
OB = b
OA = a
AP = KAB where k is a scalar quantity.
Express AB and MN in terms of a and b.
Express MP in terms of a, b and k.
Finally, find the value of k.
A

Answer 1

The value of AB can be found by using the midpoint formula, which states that the midpoint of a line segment is the average of its endpoints. Since M is the midpoint of OB, we can write the following equation to find AB: AB = (a+b)/2.

To find the value of MN, we first need to find the value of AM. Since AN is parallel to MB, we can use the following equation to find AM: AM = AN - AB = 20A - (a+b)/2. Then, we can use the midpoint formula again to find MN: MN = (AM+BM)/2 = (20A - (a+b)/2 + b)/2 = 10A - (a+b)/4.

To find the value of MP, we first need to find the value of BP. Since AP is parallel to MB and AB = k*AP, we can use the following equation to find BP: BP = AB/k = (a+b)/2k. Then, we can use the midpoint formula again to find MP: MP = (BP+BM)/2 = ((a+b)/2k + b)/2 = (a+3b)/4k.

Finally, to find the value of k, we can use the equation AB = kAP and substitute the known values for AB and AP: (a+b)/2 = k((a+b)/2k) = a/2k. Solving for k, we get: k = a/(a+b).