Question

See attachment.

Answer 1

Using Pythagoras theorem
\[AC = \sqrt{AD ^{2} + DC ^{2}} = \sqrt{900 + 1600} = \sqrt{2500} = 50\]

Answer 2

I have so many more right triangle related questions! Would you be willing to help me with as much as you can?

Answer 3

I can try....but let me finish with this one..

Answer 4

using similarity of triangles
triangle CDA is similar to triangle DBC
so

DB DC DB 40
--- = --- => ----- = ----
CD CA 40 50

so
40 * 40 1600
DB = --------- = ------ = 32
50 50

Answer 5

Triangle ADC is a right triangle with legs of 30 and 40. The hypotenuse AC = 50 because (30, 40, 50) is a Pythagorean Triple, a ten-multiple of (3, 4, 5).

Now, to find BD.

Let AB = x, then AC = 50 - x.

AC is to AD as AD is to AB

This is because of the following theorem:

If the altitude is drawn to the hypotenuse of a right triangle, the length of either leg is the geometric mean between the hypotenuse and the segment of the hypotenuse adjacent to the leg.

AC/AD = AD/AB

50/30 = 30/(50-x)

50 ( 50-x) = 30*30

x = 32 = AB.

BC = AC - AB

BC = 50 - 32 = 18.

Let y = BD.

32 is to y as y is to 18.

This is because of the theorem:

If the altitude is drawn to the hypotenuse of a right triangle, the length of the altitude is the geometric mean between the lengths of the segments of the hypotenuse.

32/y = y /18
y = 24 = BD