Class: 10th | Mathematics (FBISE) | Lecture # | Unit #11 |Theorem #4 | Chords and Arcs | скачать в хорошем качестве

Class: 10th | Mathematics (FBISE) | Lecture # | Unit #11 |Theorem #4 | Chords and Arcs |

Class 10th | Mathematics (FBISE) | Lecture # | Unit 11 | Chords and Arcs | Theorem #4 | If the angles subtended by two chords of a circle (or congruent circles) at the centre (corresponding centres) are equal, the chords are equal | The intersecting chords theorem or just the chord theorem is a statement in elementary geometry that describes a relation of the four line segments created by two intersecting chords within a circle. It states that the products of the lengths of the line segments on each chord are equal. It is Proposition 35 of Book 3 of Euclid's Elements. More precisely, for two chords AC and BD intersecting in a point S the following equation holds: The converse is true as well, that is if for two line segments AC and BD intersecting in S the equation above holds true, then their four endpoints A, B, C and D lie on a common circle. Or in other words if the diagonals of a quadrilateral ABCD intersect in S and fulfill the equation above then it is a cyclic quadrilateral. The value of the two products in the chord theorem depends only on the distance of the intersection point S from the circle's center and is called the absolute value of the power of S, more precisely it can be stated that: |AS| . |SC| = |BS| . |SD|=r^2 - d^2 where r is the radius of the circle, and d is the distance between the center of the circle and the intersection point S. This property follows directly from applying the chord theorem to a third chord going through S and the circle's center M (see drawing). The theorem can be proven using similar triangles (via the inscribed-angle theorem). Consider the angles of the triangles ASD and BSC: Next to the tangent-secant theorem and the intersecting secants theorem the intersecting chord theorem represents one of the three basic cases of a more general theorem about two intersecting lines and a circle - the power of point theorem. We will learn theorems that involve chords of a circle Perpendicular bisector of a chord passes through the center of a circle. Congruent chords are equidistant from the center of a circle. If two chords in a circle are congruent, then their intercepted arcs are congruent. If two chords in a circle are congruent, then they determine two central angles that are congruent. The following diagrams give a summary of some Chord Theorems: Perpendicular Bisector and Congruent Chords. Scroll down the page for examples, explanations, and solutions.