A chord AB of length 13 cm subtends an angle of 67° at the circumference of a circle center O. Find the radius of the circle

To find the radius of the circle, we can use the relationship between the chord length, the radius of the circle, and the angle subtended by the chord.

Step 1: Recall the relationship between the chord, radius, and central angle:
- The length of a chord is related to the radius and central angle of a circle by the formula: chord length = 2 * radius * sin(theta/2), where theta is the central angle.

Step 2: Substitute the given values:
- Chord length (AB) = 13 cm
- Central angle (theta) = 67°

Step 3: Rearrange the formula to solve for the radius:
- chord length = 2 * radius * sin(theta/2)
- radius = chord length / (2 * sin(theta/2))

Step 4: Calculate the radius using the formula:
- radius = 13 cm / (2 * sin(67°/2))

Step 5: Calculate the value of sin(67°/2):
- sin(67°/2) ≈ 0.919

Step 6: Substitute the value of sin(67°/2) in the formula to find the radius:
- radius = 13 cm / (2 * 0.919)
- radius ≈ 7.08 cm

Therefore, the radius of the circle is approximately 7.08 cm.

To find the radius of the circle, we can use the properties of a circle and the given information.

First, let's draw a diagram to visualize the problem:

```
C (center of the circle, O)
/
/ R (radius of the circle)
/
A ---- B (chord AB)
```

Given that AB is a chord of length 13 cm, we can divide it in half to find the perpendicular bisector, which passes through the center of the circle. Let's call the midpoint of AB as point M:

```
C (center of the circle, O)
/
/ R (radius of the circle)
/
A - M - B
```

Since the chord subtends an angle of 67° at the circumference, we can create a triangle OAB with angle O = 67°:

```
C (center of the circle, O)
/ |\
/ | \
/ | \
A - M - B - O
```

In triangle OAB, O is the center of the circle, and OM is the perpendicular bisector of AB. Thus, OM is also the radius of the circle.

To find the radius, we need to find the length of OM. We can use trigonometry and the relationship between a side and an angle in a right triangle.

The side adjacent to angle O is OM, and the hypotenuse is OA (which is the radius of the circle).

Using the cosine function:

cos(O) = OM / OA

Substituting the given values:

cos(67°) = OM / OA

Now, solve for OM:

OM = cos(67°) × OA

Since OM is half the length of AB (as OM is the midpoint), we can find it by dividing AB by 2:

OM = 13 cm / 2
OM = 6.5 cm

Substitute this value into the equation:

6.5 cm = cos(67°) × OA

To isolate OA, divide both sides of the equation by cos(67°):

OA = 6.5 cm / cos(67°)

Using a calculator, find the value of cos(67°) and calculate OA:

OA ≈ 13.01 cm

Therefore, the radius of the circle is approximately 13.01 cm.

I made a sketch, which you obviously have done as well.

let the radius be r, then I see that
r^2 + r^2 - 2(r)(r)cos67° = 13^2
r^2 (2 - 2cos67°) = 169
find r