Final answer:
Using the Pythagorean theorem, we find AC, the hypotenuse of the right-angled triangle ABC, to be 20 units long. To find BK, we use the similarity of triangles ABK and ABC, resulting in BK being 9.6 units long.
Step-by-step explanation:
To solve for AC and BK in triangle ABC, where angle B is 90 degrees, AB is 12, and BC is 16, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Since AB and BC are the two legs of the triangle and angle B is 90 degrees, we can find AC (the hypotenuse) by applying the theorem:
AC2 = AB2 + BC2
= 122 + 162
= 144 + 256
= 400
Hence, AC = √(400) = 20.
Now, to find BK, we note that triangle ABK is similar to triangle ABC (both are right-angled, and share a common angle at A), which means that the sides are proportional. Let BK be x, then we have:
AB / BK = AC / BC
12 / x = 20 / 16
x = (12 * 16) / 20
x = 192 / 20
x = 9.6
Thus, BK is 9.6 units long.