Question

В прямоугольном треугольнике АВС, угол В=90 градусов, АВ=8 см, АС=16 см. Найдите углы, которые образует высота ВН с катетами треугольника. ПОЖАЛУЙСТА ПОМОГИТЕ! НО МНЕ НУЖНО С ПОЛНЫМ РЕШЕНИЕМ И РИСУНКОМ, А ТАКЖЕ Я ДАЮ 30 баллов! И ОТМЕЧУ КАК ЛУЧШИЙ!

Answer 1

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Answer 2

Problem Analysis

We are given a right triangle ABC, where angle B is 90 degrees, AB is 8 cm, and AC is 16 cm. We need to find the angles formed by the altitude BH with the legs of the triangle.

Solution

To solve this problem, we can use the properties of right triangles and trigonometry.

Let's start by finding the length of the altitude BH. In a right triangle, the altitude from the right angle to the hypotenuse divides the triangle into two similar triangles. Therefore, we can use the following proportion:

BH/AB = AC/BC

Substituting the given values, we have:

BH/8 = 16/BC

Cross-multiplying, we get:

BH * BC = 8 * 16

Simplifying, we have:

BH * BC = 128

Now, let's find the angles formed by the altitude BH with the legs of the triangle. We can use trigonometric ratios to find these angles.

Let angle A be the angle formed by BH with leg AC, and angle C be the angle formed by BH with leg BC.

Using the sine ratio, we have:

sin(A) = BH/AC

Substituting the given values, we have:

sin(A) = BH/16

Rearranging the equation, we get:

BH = 16 * sin(A)

Similarly, using the sine ratio for angle C, we have:

sin(C) = BH/BC

Substituting the given values, we have:

sin(C) = BH/BC

Rearranging the equation, we get:

BH = BC * sin(C)

Since BH is the same in both equations, we can equate the two expressions for BH:

16 * sin(A) = BC * sin(C)

Now, we can solve this equation to find the values of angles A and C.

Let's calculate the values using the given information.

Calculation

Using the given information: - AB = 8 cm - AC = 16 cm

We can calculate the values of angles A and C.

Using the equation BH * BC = 128, we can find the value of BC:

BH * BC = 128

Substituting the value of BH from the equation BH = 16 * sin(A), we have:

16 * sin(A) * BC = 128

Simplifying, we get:

sin(A) * BC = 8

Now, using the equation sin(C) = BH/BC, we can find the value of sin(C):

sin(C) = BH/BC

Substituting the value of BH from the equation BH = BC * sin(C), we have:

sin(C) = BC * sin(C)/BC

Simplifying, we get:

sin(C) = sin(C)

Since sin(C) = sin(C), we can conclude that angle C is equal to angle C.

Now, let's solve the equation sin(A) * BC = 8 to find the value of BC.

Using the given information: - AB = 8 cm - AC = 16 cm

We can substitute the values into the equation:

sin(A) * BC = 8

Substituting the value of sin(A) from the equation sin(A) = BH/16, we have:

(BH/16) * BC = 8

Simplifying, we get:

BH * BC = 128

Since we already know that BH * BC = 128, we can conclude that BC = 8.

Now, let's find the values of angles A and C.

Using the equation sin(A) * BC = 8, we have:

(BH/16) * 8 = 8

Simplifying, we get:

BH = 16

Therefore, angle A is equal to the inverse sine of BH/AC:

A = sin^(-1)(16/16) = sin^(-1)(1) = 90 degrees

Similarly, angle C is equal to the inverse sine of BH/BC:

C = sin^(-1)(16/8) = sin^(-1)(2) = 90 degrees

Answer

The angles formed by the altitude BH with the legs of the triangle are both 90 degrees.

I hope this helps! Let me know if you have any further questions.