Question

В треугольнике АВС угол С=90°, угол А=30°, АВ=25. Найдите ВС

Answer 1

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Answer 2

To find the length of side BC in triangle ABC, we can use trigonometric ratios. Since angle C is a right angle (90°) and angle A is 30°, we can use the trigonometric ratio for the sine of angle A and the cosine of angle A:

1. The sine of angle A (sin A) is defined as the ratio of the side opposite angle A to the hypotenuse (the longest side) in a right-angled triangle.

   sin A = Opposite / Hypotenuse

2. The cosine of angle A (cos A) is defined as the ratio of the side adjacent to angle A to the hypotenuse in a right-angled triangle.

   cos A = Adjacent / Hypotenuse

Let's find the lengths of sides AB and BC:

Given: Angle A = 30° Angle C = 90° AB = 25 (opposite side to angle A)

1. Finding the length of side BC (hypotenuse): We can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

BC^2 = AB^2 + AC^2

Since angle C is 90°, AC is the adjacent side to angle A, and we can use the cosine of angle A to find its length.

cos A = AC / Hypotenuse cos 30° = AC / BC √3/2 = AC / BC

Now, let's substitute the values we know:

BC^2 = 25^2 + (AC)^2 BC^2 = 625 + (BC * √3/2)^2 BC^2 = 625 + 3/4 * BC^2

Now, let's solve for BC:

BC^2 - 3/4 * BC^2 = 625 (1 - 3/4) * BC^2 = 625 (1/4) * BC^2 = 625

BC^2 = 625 / (1/4) BC^2 = 625 * 4 BC^2 = 2500

BC = √2500 BC = 50

So, the length of side BC is 50 units.