Calculating Altitude of Right Triangle
The problem presents a right triangle ABC with the right angle at B and an altitude BD drawn to the hypotenuse AC. The lengths given are AD = 12 and DC = 16.
To find the length of BD in simplest radical form, we're going to make use of similar triangles. The altitude of a right triangle creates two smaller triangles (ABD and CBD) which are similar to each other and also similar to the original triangle ABC.
By the properties of similar triangles, the ratios of corresponding sides are equal. So for triangles ABD and ABC, the following ratio holds:
AD/AB = AB/AC
Given that AD is 12 and AC (the entire hypotenuse) is 12 + 16 = 28, we can say:
AB/28 = 12/AB
Solving for AB, we get the equation:
AB^2 = 12 * 28
AB^2 = 336
AB = √336
AB = √(16*21)
AB = 4√21 (since √16 = 4)
Now, let us determine the length of BD. The similar triangles ABD and ABC also give us:
BD/AB = AB/AC
Filling in the values we've found:
BD/4√21 = 4√21/28
Cross-multiplying gives:
BD = (4√21 * 4√21) / 28
BD = (16 * 21) / 28
BD = 16 * 3/4 (because 21/28 simplifies to 3/4)
BD = 12
So the length of BD, which is the altitude of the right triangle ABC, is 12.
