9514 1404 393
Answer:
A) 1
Explanation:
Let the distance from A to BC be represented by y, and the distance DE be represented by x. We can write some relations using the definition of the tangent of an angle, and the double-angle formula for tangent. The double-angles are complementary
Left angles
Call the smaller angle α. Then the larger angle is 2α.
tan(α) = x/2
tan(2α) = 2tan(α)/(1 -tan(α)²) = 2(x/2)/(1 -(x/2)²) = 4x/(4 -x²)
Right angles
tan(β) = x/3
tan(2β) = 2tan(β)/(1 -tan(β)²) = 2(x/3)/(1 -(x/3)²) = 6x/(9 -x²)
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Angles 2α and 2β are complementary, so the product of their tangents is 1.
(4x/(4 -x²))(6x/(9 -x²)) = 1
24x² = (4 -x²)(9 -x²) . . . . multiply by the product of denominators
x⁴ -37x² +36 = 0 . . . . . put in standard form
(x² -36)(x² -1) = 0 . . . . . factor
Solutions to this equation are x = ±1 and x = ±6. Only positive values less than 1.25 can possibly satisfy the geometry. Hence x = DE = 1.
The length of DE is 1.