https://www.khanacademy.org/math/geometry-tx/x790e3ac3e338c450:relationships-of-circles-including-radian-measure-and-equations-of-circles/x790e3ac3e338c450:properties-of-chords/v/proof-the-product-of-the-segments-of-intersecting-chords 2025-05-12T16:49:01.249479039Z Proof The product of the segments of intersecting chords Let's prove the general case that the product of the segments of intersecting chords are equal. We'll create similar triangles and use the properties of inscribed angles and vertical angles to convince ourselves that this is always true. Sal Khan video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Proof The product of the segments of intersecting chords Let's prove the general case that the product of the segments of intersecting chords are equal. We'll create similar triangles and use the properties of inscribed angles and vertical angles to convince ourselves that this is always true. https://cdn.kastatic.org/ka-youtube-converted/3hV7ZXxal7k.mp4/3hV7ZXxal7k.mp4 284 Properties of chords https://www.khanacademy.org/math/geometry-tx/x790e3ac3e338c450:relationships-of-circles-including-radian-measure-and-equations-of-circles/x790e3ac3e338c450:properties-of-chords/v/proof-angle-formed-by-intersecting-chords 2025-05-12T16:49:01.249479039Z Proof: Angle formed by intersecting chords Let's prove that the angle formed by intersecting chords is half the sum of the measures of the intercepted arcs. We'll construct a triangle, then use the properties of inscribed angles to convince ourselves that this is always true. Sal Khan video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Proof: Angle formed by intersecting chords Let's prove that the angle formed by intersecting chords is half the sum of the measures of the intercepted arcs. We'll construct a triangle, then use the properties of inscribed angles to convince ourselves that this is always true. https://cdn.kastatic.org/ka-youtube-converted/nxHUx1o8-sA.mp4/nxHUx1o8-sA.mp4 270 Properties of chords https://www.khanacademy.org/math/cambridge-o-level-mathematics-4024/x3f4c73d082a798eb:geometry/x3f4c73d082a798eb:circle-theorem-i/v/proof-angle-between-a-tangent-and-a-chord 2025-05-12T16:49:01.249479039Z Proof: Angle between a tangent and a chord Let's prove that the measure of the angle formed by the intersection of a tangent and a chord is half the measure of the intercepted arc. We'll construct an isosceles triangle and use the relationship between radii and tangents of a circle to convince ourselves that this is true for both the major and minor arcs formed. Sal Khan video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Proof: Angle between a tangent and a chord Let's prove that the measure of the angle formed by the intersection of a tangent and a chord is half the measure of the intercepted arc. We'll construct an isosceles triangle and use the relationship between radii and tangents of a circle to convince ourselves that this is true for both the major and minor arcs formed. https://cdn.kastatic.org/ka-youtube-converted/hbdvuygTlMA.mp4/hbdvuygTlMA.mp4 328 Properties of chords https://www.khanacademy.org/math/geometry-tx/x790e3ac3e338c450:relationships-of-circles-including-radian-measure-and-equations-of-circles/x790e3ac3e338c450:properties-of-chords/v/proof-diameter-that-bisects-a-chord 2025-05-12T16:49:01.249479039Z Proof: Diameter that bisects a chord Let's prove that if a diameter of a circle bisects a chord, it is perpendicular to the chord. We'll construct congruent triangles to convince ourselves that this is always true. Sal Khan video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Proof: Diameter that bisects a chord Let's prove that if a diameter of a circle bisects a chord, it is perpendicular to the chord. We'll construct congruent triangles to convince ourselves that this is always true. https://cdn.kastatic.org/ka-youtube-converted/diQqp7pWxZY.mp4/diQqp7pWxZY.mp4 124 Properties of chords https://www.khanacademy.org/math/cambridge-o-level-mathematics-4024/x3f4c73d082a798eb:geometry/x3f4c73d082a798eb:circle-theorem-ii/e/chords-in-circles 2025-06-18T23:30:37.717264784Z Chords in circles Given 2 chords that intersect within a circle and some relevant measures, calculate a missing measurement. Charlie Auen exercise