In angle ABC, AB = 8 cm, BC = 4 cm, CA = 5 cm and BC is produced to P so that CP = 4 cm. Use the cosine rule to find Cos ACB. Hence, find AP

c^2 = a^2 + b^2 - 2ab cos C

8^2 = 4^2 + 5^2 - 2*4*5 cosC
C = 125°
So, angle ACP = 55°
AP^2 = 4^2 + 5^2 - 2*4*5 cos55°
AP = 4.25

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To find the value of cos(ACB) using the cosine rule, we need to determine the lengths of the three sides of triangle ABC.

From the given information, we have:
AB = 8 cm
BC = 4 cm
CA = 5 cm

Let's label the extension of side BC as CP, where CP = 4 cm.

To find the value of cos(ACB), we first need to find the value of angle ACB itself.

Using the cosine rule, we have:
cos(ACB) = (AB^2 + BC^2 - CA^2) / (2 * AB * BC)

Plugging in the given values, we get:
cos(ACB) = (8^2 + 4^2 - 5^2) / (2 * 8 * 4)
= (64 + 16 - 25) / 64
= 55 / 64

So, cos(ACB) = 55 / 64.

Now, to find AP, we know that CP is the extension of side BC. Since we are given that CP = 4 cm, we can find AP by subtracting CP from the length of side BC.

AP = BC - CP
= 4 cm - 4 cm
= 0 cm

Hence, AP = 0 cm.