Question

Трикутник АВС прямокутний кут А=60 кут С=90 АТ-бісектриса кута А ТВ= 12см Знайти СВ

Answer 1

Ответ: СВ = 18 см

Объяснение:

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Answer 2

Triangle ABC with Right Angle

We have a triangle ABC with a right angle at angle C. Angle A is 60 degrees and angle C is 90 degrees. The bisector of angle A is AT, and TV is 12 cm. We need to find the length of SV.

To solve this problem, we can use the properties of right-angled triangles and the angle bisector theorem.

Angle Bisector Theorem

The angle bisector theorem states that in a triangle, the length of the side opposite to the angle bisector is proportional to the lengths of the other two sides. Mathematically, we can express this as:

AB / AC = TB / TC

In our case, we know that AB = 12 cm (given as TV) and AC = x (unknown). We also know that TB = x (since AT is the bisector of angle A) and TC = SV (unknown). Plugging these values into the angle bisector theorem equation, we get:

12 / x = x / SV

Solving for SV

To find the value of SV, we can rearrange the equation obtained from the angle bisector theorem:

12 * SV = x^2

Since we don't have the exact value of x, we need to eliminate it from the equation. To do this, we can use the fact that angle A is 60 degrees. In a right-angled triangle, the sum of the other two angles is always 90 degrees. Therefore, angle B is 30 degrees.

Using the trigonometric ratios for a right-angled triangle, we can express the relationship between the sides and angles as follows:

sin(30) = SV / x

Since sin(30) = 1/2, we can rewrite the equation as:

1/2 = SV / x

From this equation, we can solve for x:

x = 2 * SV

Substituting this value of x into the equation obtained from the angle bisector theorem, we get:

12 * SV = (2 * SV)^2

Simplifying this equation, we have:

12 * SV = 4 * SV^2

Dividing both sides of the equation by SV, we get:

12 = 4 * SV

Finally, solving for SV, we have:

SV = 12 / 4 = 3 cm

Therefore, the length of SV is 3 cm.

Summary

In summary, we have a right-angled triangle ABC with angle A measuring 60 degrees and angle C measuring 90 degrees. AT is the bisector of angle A, and TV is given as 12 cm. Using the angle bisector theorem and trigonometric ratios, we can determine that the length of SV is 3 cm.