If two chords in a circle are equal in length, they are equidistant from the center of the circle.

The intersecting chords theorem states that the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.

The area ratio is the square of the radius ratio. If area ratio = 1:4, then radius ratio = 1:2. Therefore, the radius of the second circle = 4 * 2 = 8 cm.

In similar circles, the radii are proportional.

Area ratio = (r1/r2)² = (3/5)² = 9/25.

The distance between the centers of two externally tangent circles is the sum of their radii: 3 cm + 4 cm = 7 cm.

Distance between centers = r1 + r2 = 3 + 5 = 8 cm.

The area of a circle is A = πr². The areas are 9π cm² and 25π cm², so the ratio of their areas is 9:25.

Area ratio = (r1²):(r2²) = (3²):(6²) = 9:36 = 1:4.

The distance between the centers of two externally tangent circles is the sum of their radii: 3 + 4 = 7 units.

Area ratio = (r1^2):(r2^2) = (4^2):(6^2) = 16:36 = 4:9.

The perpendicular from the center of a circle to a chord bisects the chord.

The angles subtended by the common chord AB at the centers of the circles are equal.

The line segment AB is a chord of both circles since it connects two points on each circle.

The angles subtended by the same arc at the circumference are equal, hence angle ACB = angle APB.

The line joining the two points of intersection of two circles is called a chord.

Two circles can intersect at two points regardless of whether their radii are equal or different.

For independent events, P(A and B) = P(A) * P(B).

If alternate exterior angles are equal, the lines are parallel.

Consecutive interior angles are supplementary when two lines are parallel. Therefore, 110 + x = 180, so x = 70 degrees.

If the corresponding angles are equal, then the lines are parallel according to the Corresponding Angles Postulate.

The interior angle on the same side is supplementary to the exterior angle, so it measures 180 - 130 = 50 degrees.

If the interior angles are supplementary, it indicates that the lines are not parallel, as parallel lines would create equal alternate interior angles.

Corresponding angles are equal, so the corresponding angle is also 120 degrees.

Corresponding angles are equal, so the corresponding angle is also 40°.

The alternate exterior angle is equal to the angle formed by the transversal, so it is 40°.

Alternate interior angles are equal, so x is 75 degrees.

Vertically opposite angles are equal. Therefore, the vertically opposite angle also measures 75 degrees.

Corresponding angles are equal when a transversal intersects parallel lines, so the corresponding angle is also 30 degrees.

The angles formed are supplementary because they are same-side interior angles. Thus, 120 + x = 180, leading to x = 60 degrees.