Question

Радиус OB окружности с центром в точке O пересекает хорду AC в точке D и перпендикулярен ей. Найдите длину хорды AC , если BD = 2 см, а радиус окружности равен 5 см

Answer 1

ОД=ОВ-ДВ=3

проведём радиус ОА и найдём длину АД по теореме пифагора АД^2=ОА^2-ОД^2

АД^2=5^2-3^2=25-9=16 следовательно АД=4  аналогично найдём ДС, ДС^2=5^2-3^2=25-9=16 следовательно ДС=4. АС=4+4=8

Answer 2

Understanding the problem

We have a circle with center O and radius OB. The circle intersects chord AC at point D, and BD is perpendicular to AC. We need to find the length of chord AC given that BD is 2 cm and the radius of the circle is 5 cm.

Solution

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

Let's start by drawing a diagram to visualize the problem:

          O
         /|\
        / | \
     B /  |  \ A
      /   |   \
     /    |    \
    /     |     \
   D------C------

From the problem statement, we know that BD is perpendicular to AC. This means that triangle BCD is a right triangle. Let's label the length of BC as x.

Using the Pythagorean theorem, we can find the length of BD:

BD^2 = BC^2 + CD^2

Since BD = 2 cm, we can substitute the values:

2^2 = x^2 + CD^2

4 = x^2 + CD^2

Now, let's look at triangle BOC. OB is the radius of the circle, which is given as 5 cm. Since OB is the radius, it is also equal to BC.

Since triangle BOC is isosceles (with two sides of equal length), we know that angle BOC is also equal to angle BCO. Therefore, angle BOC is a right angle.

Since angle BOC is a right angle, triangle BOC is also a right triangle.

Now, we can use the Pythagorean theorem to find the length of BC:

BC^2 = OB^2 + OC^2

BC^2 = 5^2 + OC^2

BC^2 = 25 + OC^2

Since BC = x, we can substitute the values:

x^2 = 25 + OC^2

Now, let's combine the equations we have:

4 = x^2 + CD^2

x^2 = 25 + OC^2

We can rewrite the second equation as:

OC^2 = x^2 - 25

Now, we can substitute this into the first equation:

4 = x^2 + CD^2

4 = x^2 + (x^2 - 25)

4 = 2x^2 - 25

29 = 2x^2

x^2 = 29/2

x = sqrt(29/2)

Therefore, the length of chord AC is 2 times the length of BC:

AC = 2 * sqrt(29/2)

Simplifying this expression, we get:

AC = sqrt(58)

So, the length of chord AC is sqrt(58) cm.