Hypotenuse 64694

Point S is the center of the hypotenuse AB of the right triangle ABC. Calculate the content of triangle ABC if the line on the hypotenuse is 0.2 dm long and if | ∢ACS | = 30 °.

Correct answer:

S = 3.4641 cm²

Step-by-step explanation:

t = 0.2 dm \to cm = 0.2 \cdot 10 cm = 2 cm ∠ A C S = 30^{\circ} ∠ S C B = 90 - ∠ A C S = 90 - 30 = 60^{\circ} \frac{sin A C S}{sin α} = \frac{c / 2}{t} \frac{sin S C B}{sin β} = \frac{c / 2}{t} β = 90 ° - α \frac{sin A C S}{sin α} = \frac{sin S C B}{sin ( 9 0 ° - α )} sin A C S \cdot sin (90 ° - α) = sin S C B \cdot sin α sin (α - β) = sin α . cos β - cos α . sin β sin A C S \cdot (sin 90 ° cos α - cos 90 ° sin α) = sin S C B \cdot sin α sin A C S \cdot cos α = sin S C B \cdot sin α sin A C S / sin S C B = sin α / cos α tan α = sin A C S / sin S C B α = \frac{1 8 0 °}{π} \cdot arctan (sin A C S / sin S C B) = \frac{1 8 0 °}{π} \cdot arctan (sin 30 ° / sin 60 °) = \frac{1 8 0 °}{π} \cdot arctan (0.5 / 0.866025) = 30 β = 90 - α = 90 - 30 = 60^{\circ} c = 2 \cdot t \cdot \frac{sin A C S}{sin α} = 2 \cdot t \cdot \frac{sin 3 0 °}{sin 3 0 °} = 2 \cdot 2 \cdot \frac{sin 3 0 °}{sin 3 0 °} = 2 \cdot 2 \cdot \frac{0 . 5}{0 . 5} = 4 cm a = c \cdot sin α = c \cdot sin 30 ° = 4 \cdot sin 30 ° = 4 \cdot 0.5 = 2 cm b = c \cdot sin β = c \cdot sin 60 ° = 4 \cdot sin 60 ° = 4 \cdot 0.866025 = 3.4641 cm S = \frac{a \cdot b}{2} = \frac{2 \cdot 3 . 4 6 4 1}{2} = 23 = 3.4641 cm^{2}

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