help fast (10 min)

Question

In triangle ABC, ZA = 90°, and let D be the foot of the perpendicular from A to BC. If BD = 2

and DC = 8, find the length of AD.

Answer 1

This will be a little hard to answer without actually drawing the triangle but fill free to reach out to me if you still need help understanding how I derived my formulas:

The trick with the question is you have 3 unknowns, which means you need 3 equations to solve for 3 unknowns. Those unknowns are lengths AB, AC, and AD (which is what the question is asking for).

If we let AB = x & AC = y, we can use Pythagorean theorem to derive our first equation:

AB² + AC² = BC²

we know that BC = BD + DC = 2+8 = 10. Rewrite this in terms of x and y, we get:

x² + y² = 10² ... (1) [Equation 1]

The problem tells you that there's another line that splits the triangle into two. The line is AD. Now here's the key: the two triangles share AD as a side. Let's label AD as "z". Because AD is perpendicular to BC, we can use Pythagorean theorem again and generate two more equations, one for each smaller triangle:

For smaller triangle 1:

AD² + 2² = AB² or z² + 4 = x² ... (2) [Equation 2]

For smaller triangle 2:

AD² + 8² = AC² or z² + 64 = y² ... (3) [Equation 3]

So now we have our three equations and we have three unknowns. So solve for z, which is length of AD:

The three equations are:

x² + y² = 10² ... (1)

z² + 4 = x² ... (2)

z² + 64 = y² ....(3)

Substitute (2) & (3) into (1) ... and you get

(z² + 4) + (z² + 64) = 10²

2z² + 68 = 100

2z² = 32

z² = 16

z = 4

Since z = AD, your final answer is: AD = 4