Question

Пожалуйста, помогите! Умоляю! Решил половину, но дальше СТОП. В треугольнике ABC углы относятся как 2:5:3 (Угол А - 2, угол В - 5, угол С - 3). Из вершины бОльшего угла В опущен перпендикуляр BE на противоположную сторону АС. Точка D находится на стороне ВС так, что DE=BD. Найти угол EDC. Углы нашёл, но почему то угол В равен 90! Но чисто логически этого быть не может в самом обычном треугольнике.

Answer 1

Problem Analysis

We are given a triangle ABC with angles A, B, and C in the ratio 2:5:3. From the vertex of the larger angle B, a perpendicular BE is dropped onto the opposite side AC. Point D lies on side BC such that DE = BD. We need to find the angle EDC.

Solution

To find the angle EDC, we need to analyze the given information and make use of geometric properties.

Let's start by finding the measures of angles A, B, and C. We are given that the angles are in the ratio 2:5:3. We can assign variables to these angles as follows: - Angle A = 2x - Angle B = 5x - Angle C = 3x

Since the sum of the angles in a triangle is always 180 degrees, we can write the equation: 2x + 5x + 3x = 180

Simplifying the equation, we get: 10x = 180 x = 18

Now we can find the measures of angles A, B, and C: - Angle A = 2x = 2 * 18 = 36 degrees - Angle B = 5x = 5 * 18 = 90 degrees - Angle C = 3x = 3 * 18 = 54 degrees

Next, let's analyze the triangle and the perpendicular BE. Since BE is perpendicular to AC, angle BEC is a right angle (90 degrees). This means that angle B + angle BEC = 90 degrees.

Given that angle B is 90 degrees, we can conclude that angle BEC is also 90 degrees. Therefore, triangle BEC is a right triangle.

Now, let's consider the point D on side BC such that DE = BD. Since DE = BD, triangle BDE is an isosceles triangle. This means that angles BDE and BED are equal.

Since angle BDE = angle BED and angle BEC = 90 degrees, we can conclude that angle EDC is also 90 degrees. Therefore, the measure of angle EDC is 90 degrees.

In summary, the measure of angle EDC is 90 degrees.

Conclusion

The measure of angle EDC is 90 degrees. This can be deduced from the given information and the properties of triangles and perpendicular lines.

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