In a rectangular ∆ABS, the angle is C = 90 degrees, CD is the height, and one of the legs is twice as large as the other.

In a rectangular ∆ABS, the angle is C = 90 degrees, CD is the height, and one of the legs is twice as large as the other. The bisectors DK and DP are drawn in ∆ACD and ∆BCD, respectively. Find the area ∆ABS if KP = 4.

The height CD of a right-angled triangle ⊿ ABC divides it into similar triangles:
⊿ АСD ~ ⊿ ВСD = СD: ВD = 2: 1.
Because DK and DP are bisectors, then ∠ ADK = ∠ KDC = ∠ CDP = ∠ BDP = 450 ⇒ ∠ КDP = 90º.
The quadrangle КCDP is a square, therefore СD = KP = 4. By the Pythagorean theorem we find СP = √16 = 2 √2 /
The bisector of a triangle divides a side into segments proportional to the adjacent sides ⇒
СР: РВ = 2: 1⇒
2 РВ = 2 √2;
PB = √2;
BC = CP + PB = 2√2 + √2 = 3√2;
AC = 2 BC = 6√2;
S⊿ ABC = AC * BC: 2 = (6√2 * 3√2): 2 = 18.
Answer: 18