Final answer:
Side AB in triangle ABC with a 60-degree angle at A and a height of 8 from B to AC is found using the sine function, yielding AB ≈ 9.24 units.
Step-by-step explanation:
To find the length of side AB in triangle ABC with a 60-degree angle at A and a perpendicular height of 8 from B to AC, we can use trigonometry. Specifically, we will use the sine function, which relates the ratio of the opposite side to the hypotenuse in a right-angle triangle. In this case, since the height from B to AC is perpendicular, triangle ABC is divided into two right-angled triangles with angle A being 60 degrees in one of them.
The sine of angle A (60 degrees) is equal to the opposite side (height from B to AC, which is 8) divided by the hypotenuse (AB). Hence:
sin(60°) = Opposite / Hypotenuse
sin(60°) = 8 / AB
Since sin(60°) is √3/2, we have:
√3/2 = 8 / AB
Therefore, AB can be calculated as follows:
AB = 8 / (√3/2)
AB = 8 × 2 / √3
AB = 16 / √3
Finally, rationalizing the denominator, we get:
AB = 16√3 / 3
AB ≈ 9.24 (to two decimal places)
Thus, the length of side AB in triangle ABC is approximately 9.24 units.