In a triangle ABC, AB=8cm, BC=4cm, CA=5cm and BC is produced to P so that CP=4cm. Use cosine rule to find cos ACB. Hence find AP.
To find cos ACB using the cosine rule, we can use the formula:
cos ACB = (AB^2 + BC^2 - CA^2) / (2 * AB * BC)
Plugging in the values, we have:
cos ACB = (8^2 + 4^2 - 5^2) / (2 * 8 * 4)
= (64 + 16 - 25) / 64
= 55 / 64
To find AP, we can use the fact that cos ACB = cos APC, where PC is the extension of BC.
Since cos ACB = 55/64, we have:
cos APC = 55/64
Using the cosine rule again, we can write:
AP^2 = BC^2 + PC^2 - 2 * BC * PC * cos APC
Substituting the given values, we have:
AP^2 = 4^2 + 4^2 - 2 * 4 * 4 * (55/64)
= 16 + 16 - 2 * 4 * 4 * (55/64)
= 32 - 35
= -3
Since the value of AP^2 is negative, we can conclude that there is no real solution for AP.