a chord in a circle makes an angle of 45° with the radius. if the chord is 4 cm long, what is the radius of the circle

since the chord forms the base of an isosceles triangle, the other angles are 45° and 90°. The radii form the 90° angle, so 2r^2 = 4^2, and r=2√2.

To find the radius of the circle, we can use the properties of a right-angled triangle formed by the radius, the chord, and a line segment that bisects the chord.

Let's denote the radius of the circle as 'r'. We know that the chord is 4 cm long, and it makes an angle of 45° with the radius.

First, let's draw the circle and the chord. Then, draw a line from the center of the circle to the midpoint of the chord so that it bisects the chord. This line will be perpendicular to the chord, forming a right-angled triangle.

Next, let's label the relevant points. The center of the circle will be labeled 'O', the midpoint of the chord will be labeled 'M', and the endpoints of the chord will be labeled 'A' and 'B'.

The right-angled triangle is formed by the radius (OM), half the length of the chord (MA), and the line segment from O to A or B (OA or OB).

Since MA is half the length of the chord, which is 4 cm, MA = 2 cm.

Now, we can use the trigonometric ratio for the tangent of an angle to find the value of r.

In the right-angled triangle, the tangent of the angle between the radius and the chord can be defined as:

tan(45°) = MA / OA

Rearranging the equation, we have:

OA = MA / tan(45°)
OA = 2 cm / tan(45°)

The tangent of 45° is equal to 1, so we have:

OA = 2 cm / 1
OA = 2 cm

Therefore, the length of OA (which is also the radius, denoted as 'r') is 2 cm. Hence, the radius of the circle is 2 cm.