The diagram shows a trapezium ABCD. AB is parallel to DC. AB=6a, DC=2a and DA=3b. AC and BD intersect at P such that AP:PC=3:1. Express in terms of a and/ or b
AC
Since AP:PC = 3:1, we can express AP as 3x and PC as x for some value of x.
From triangle ADC, we can use the triangle proportionality theorem to find that AC/CD = AP/PD. Since AB is parallel to CD, triangle ADC is similar to triangle ABC. Therefore, we can replace AD with AB in the proportion.
We get AC/2a = 3x/(3x + 6a).
Cross multiplying, we have AC(3x + 6a) = 2a * 3x.
Expanding and simplifying, we have 3ACx + 6ACa = 6ax.
Dividing both sides by 3 and factoring out an x, we have ACx + 2ACa = 2ax.
Dividing both sides by x, we have AC + 2aAC/x = 2a.
Since AP is 3x, we can replace x with AP/3 to get AC + 2aAC/(AP/3) = 2a.
Multiplying both sides by 3/1, we have 3AC + 6aAC/AP = 6a.
Dividing both sides by 3, we have AC + 2aAC/AP = 2a.
Multiplying both sides by AP, we have AC(AP) + 2aAC = 2a(AP).
Using the fact that AP:PC = 3:1, we can express AP as 3(PC).
Therefore, we have AC(3PC) + 2aAC = 2a(3PC).
Expanding, we have 3ACPC + 2aAC = 6aPC.
Rearranging, we have 3ACPC = 6aPC - 2aAC.
Factoring out a 2, we have 3ACPC = 2(3aPC - aAC).
Dividing both sides by 3a, we have ACPC/a = 2(3PC - AC).
Therefore, AC = a(2(3PC - AC)/PC).
To find the length of AC in terms of a and b, we can use the concept of similar triangles.
Since AB is parallel to DC, triangle ABD and triangle DCP are similar. Therefore, we can set up the following proportion:
AD/DC = AB/CP
Given that AD = 3b, DC = 2a, and AB = 6a, we can substitute these values into the proportion:
(3b)/(2a) = (6a)/CP
Now, we can cross-multiply:
3b * CP = 2a * 6a
3b * CP = 12a^2
Next, divide both sides of the equation by 3b:
CP = (12a^2)/(3b)
Simplifying the expression gives:
CP = 4a^2/b
Since AP:PC = 3:1, we can express AP in terms of CP:
AP = 3 * CP
Substituting the value of CP, we get:
AP = 3 * (4a^2/b)
Simplifying further gives:
AP = (12a^2)/b
To find AC, we can add AP and PC:
AC = AP + PC
AC = (12a^2)/b + 4a^2/b
Finally, we can factor out a common term:
AC = (12a^2 + 4a^2)/b
AC = (16a^2)/b
Therefore, the expression for AC in terms of a and b is (16a^2)/b.