In a right-angled triangle ABC with a right angle C, the outside angle at the vertex A is 120

| September 25, 2021 | education

In a right-angled triangle ABC with a right angle C, the outside angle at the vertex A is 120 °, AC + AB = 18 cm. Find AC and AB.

1) According to the conditions of the problem, the outer angle A is equal to 120 ° degrees. And since the sum of the outer and inner angles is 180 °, it follows from this: the inner angle is A = 180 ° -120 ° = 60 °.
2) The sum of the angles of a triangle is always 180 °. We know two angles: angle C is 90 ° and angle A = 60 °. Thus, the angle B = 180 ° – (60 ° + 90 °) = 30 °.
3) As you know, sin 30 ° = 1/2. But also sin 30 ° = AC / AB. It follows from this that: AC / AB = 1/2; AB = 2 * AC.
4) According to the specified data in the conditions of the problem: AC + AB = 18 cm. We draw up an equation in which instead of AB we substitute 2 * AC (see point 3): AC + 2 * AC = 18;
5) We solve this equation: 3 * AC = 18; AC = 6 (cm).
6) AB = 18 – 6 = 12 (cm).
Answer: AC = 6 cm; AB = 12 cm.



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