Ina triangle ABC, AB=8cm, BC=4cm, CA=5cm and BC is produced to P so that CP=4cm. Use the cosine rule to find cos ACB. Hence find AP.
We can use the cosine rule to find the cosine of angle ACB.
According to the cosine rule, in a triangle ABC:
c^2 = a^2 + b^2 - 2abcos(C)
where a, b, and c are the lengths of the sides of the triangle opposite to angles A, B, and C, respectively.
In this case, we want to find cos ACB, so a = 8cm, b = 4cm, and c = 5cm (BC is opposing to angle ACB).
Plugging in the values into the formula:
5^2 = 8^2 + 4^2 - 2 * 8 * 4 * cos(ACB)
25 = 64 + 16 - 64cos(ACB)
25 = 80 - 64cos(ACB)
64cos(ACB) = 80 - 25
64cos(ACB) = 55
cos(ACB) = 55/64
Now, to find AP, we can use the cosine rule again in triangle ACP:
AP^2 = AC^2 + CP^2 - 2 * AC * CP * cos(ACB)
Plugging in the values:
AP^2 = 5^2 + 4^2 - 2 * 5 * 4 * (55/64)
AP^2 = 25 + 16 - 50 * (55/64)
AP^2 = 41 - (550/64)
AP^2 = (64 * 41 - 550)/64
AP^2 = (2624 - 550)/64
AP^2 = 2074/64
AP^2 = 32.46875
AP ≈ √32.46875
AP ≈ 5.7 cm
Therefore, cos ACB ≈ 55/64 and AP ≈ 5.7 cm.