The answer is E.
EXPLANATION:We need to find area of BDC. Hence, we need to find BD, DC, and BC. Or BC and altitude of BDC (which is a perpendicular dropped from D towards BC). Let's analyse the options:
From A:We know BD = 6. But we have no idea in what proportion it divides AC. Infact, we have no clue about AC. So we can't find the area of BDC. Eliminate AD.
From B:We know about AC = 20. If DC = x, then AD = 20-x. We still don't have value of BD, which is needed to get the value of BC. So we can't find the area of BDC. Eliminate B.
From both A and B:We know, BD = 6, AC = 20. If DC = x, then AD = 20-x.
Now, from Pythagoras theorem:
\( AB^2 = BD^2 + AD^2 \)
Substituting values:
\( AB^2 = 36 + (20-x)^2 \)
Similarly, \( BC^2 = BD^2 + DC^2 \)
Substituting values:
\( BC^2 = 36 + x^2 \)
Now, let's combine it to the bigger triangle.
Using pythagoras theorem again:
\( AB^2 + BC^2 = AC^2 \)
Substituting values that we got earlier: (Note: AC = 20, given).
\( 36 + (20-x)^2 + 36 + x^2 = 400 \)
\( 72 + x^2 + 400 - 40x + x^2 = 400 \)
\( 2x^2 - 40x + 72 = 0 \)
\( x^2 - 20x + 36 = 0 \)
We get two values of X = 18, and X = 2.
Hence, we get two values of DC. Either DC = 2, or DC = 18.
From both of these values, we get two different lengths of BC.
If DC = 2, \( BC = \sqrt{ 36 + 4 } \), which equals approx 6.3.
If DC = 18, \( BC = \sqrt{ 36 + 324 } \), which equals approx 19.
Hence, we would get two different area of triangle. Therefore, the answer is E.
ALTERNATE APPROACH:Quote:
Right Triangle Altitude Theorem: The measure of the altitude drawn from the vertex of the right angle of a right triangle to its hypotenuse is the geometric mean between the measures of the two segments of the hypotenuse.
This means, \( BD^2 = AD*DC \)
So, 36 = x*(20-x) (assuming DC = x, and AD = 20-x).
If we solve for x, we will again get two values of x, as 18 and 2, and we would get to the same conclusion as before.
Hence, the answer is E.