圆与圆周率Step

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[CC] Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

An inscribed angle is an angle with its vertex is the circle and its sides contain chords. The intercepted arc is the arc that is on the interior of the inscribed angle and whose endpoints are on the angle. The vertex of an inscribed angle can be anywhere on the circle as long as its sides intersect the circle to form an intercepted arc.

Inscribed Angle Theorem The measure of an inscribed angle is half the measure of its intercepted arc. To prove the Inscribed Angle Theorem, you would need to split it up into three cases, like the three different angles drawn from the Investigation.

Congruent Inscribed Angle Theorem Inscribed angles that intercept the same arc are congruent.

Inscribed Angle Semicircle Theorem An angle that intercepts a semicircle is a right angle.

In the Inscribed Angle Semicircle Theorem we could also say that the angle is inscribed in a semicircle. Anytime a right angle is inscribed in a circle, the endpoints of the angle are the endpoints of a diameter. Therefore, the converse of the Inscribed Angle Semicircle Theorem is also true.

When an angle is on a circle, the vertex is on the circumference of the circle. One type of angle on a circle is one formed by a tangent and a chord.

Chord/Tangent Angle Theorem The measure of an angle formed by a chord and a tangent that intersect on the circle is half the measure of the intercepted arc.

From the Chord/Tangent Angle Theorem, we now know that there are two types of angles that are half the measure of the intercepted arc; an inscribed angle and an angle formed by a chord and a tangent. Therefore, any angle with its vertex on a circle will be half the measure of the intercepted arc.

An angle is considered inside a circle when the vertex is somewhere inside the circle, but not on the center. All angles inside a circle are formed by two intersecting chords.

Intersecting Chords Angle Theorem The measure of the angle formed by two chords that intersect inside a circle is the average of the measure of the intercepted arcs.

An angle is considered to be outside a circle if the vertex of the angle is outside the circle and the sides are tangents or secants. There are three types of angles that are outside a circle: an angle formed by two tangents, an angle formed by a tangent and a secant, and an angle formed by two secants. Just like an angle inside or on a circle, an angle outside a circle has a specific formula, involving the intercepted arcs.

Outside Angle Theorem The measure of an angle formed by two secants, two tangents, or a secant and a tangent drawn from a point outside the circle is equal to half the difference of the measures of the intercepted arcs.

When two chords intersect inside a circle, the two triangles they create are similar, making the sides of each triangle in proportion with each other. If we remove AD and BC the ratios between AE, EC, DE, and EB will still be the same.

Intersecting Chords Theorem If two chords intersect inside a circle so that one is divided into segments of length a and b and the other into segments of length c and d then ab=cd. In other words, the product of the segments of one chord is equal to the product of segments of the second chord.

In addition to forming an angle outside of a circle, the circle can divide the secants into segments that are proportional with each other.

If we draw in the intersecting chords, we will have two similar triangles.

From the inscribed angles and the Reflexive Property (∠R≅∠R),△PRS∼△TRQ. Because the two triangles are similar, we can set up a proportion between the corresponding sides. Then, cross-multiply. ac+d=ca+b⇒a(a+b)=c(c+d)

Two Secants Segments Theorem If two secants are drawn from a common point outside a circle and the segments are labeled as above, then a(a+b)=c(c+d). In other words, the product of the outer segment and the whole of one secant is equal to the product of the outer segment and the whole of the other secant.

If a tangent and secant meet at a common point outside a circle, the segments created have a similar relationship to that of two secant rays. Recall that the product of the outer portion of a secant and the whole is equal to the same of the other secant. If one of these segments is a tangent, it will still be the product of the outer portion and the whole. However, for a tangent line, the outer portion and the whole are equal.

Tangent Secant Segment Theorem If a tangent and a secant are drawn from a common point outside the circle (and the segments are labeled like the picture to the left), then a2=b(b+c). This means that the product of the outside segment of the secant and the whole is equal to the square of the tangent segment.