Question

В треугольнике АВС сторона АС = 30, а угол В = 45 градусов. Найти радиус описанной окружности.

Answer 1

R=30/2*корень из 2 разладить на 2=15 корень из 2

Answer 2

Problem Analysis

We are given a triangle ABC with side AC measuring 30 units and angle B measuring 45 degrees. We need to find the radius of the circumcircle of triangle ABC.

Solution

To find the radius of the circumcircle, we can use the formula:

R = (abc) / (4A)

where R is the radius of the circumcircle, a, b, and c are the side lengths of the triangle, and A is the area of the triangle.

To find the area of the triangle, we can use Heron's formula:

A = sqrt(s(s-a)(s-b)(s-c))

where s is the semiperimeter of the triangle, given by:

s = (a + b + c) / 2

Let's calculate the radius of the circumcircle step by step.

1. Calculate the length of side AB using the Law of Sines:

AB / sin(B) = AC / sin(A)

Since angle A is 180 - 45 - 90 = 45 degrees, we have:

AB / sin(45) = 30 / sin(45)

Solving for AB, we get:

AB = 30

2. Calculate the length of side BC using the Law of Cosines:

BC^2 = AB^2 + AC^2 - 2 * AB * AC * cos(B)

Substituting the known values, we have:

BC^2 = 30^2 + 30^2 - 2 * 30 * 30 * cos(45)

Solving for BC, we get:

BC = 30 * sqrt(2)

3. Calculate the length of side AC using the Law of Sines:

AC / sin(C) = BC / sin(B)

Since angle C is 180 - 45 - 90 = 45 degrees, we have:

AC / sin(45) = BC / sin(45)

Solving for AC, we get:

AC = BC = 30 * sqrt(2)