Unit 4.3 Identifying, Describing, and Applying Theorems about Circles

Understand and apply theorems about circles G-C Understand and apply theorems about circles G-C.2, 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. G-C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Expressing Geometric Properties with Equations G-GPE Use coordinates to prove simple geometric theorems algebraically [Include distance formula; relate to Pythagorean theorem] G-GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). Congruence G-CO Make geometric constructions [Formalize and explain processes] G-CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

Standards of Math Practice SMP 4 Model with Mathematics Construct, inscribe, and circumscribe polygons SMP 5 Use Appropriate Tools Strategically Use technology and construction tools appropriately 

Essential Questions What is the relationship between a radii and a chord? Where would you see each part of a circle in the real world?

Identifying and Describing, Parts of Circles G-C.2 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.

Review Concepts QB is a radius A circle is a set of points in a plane that are the same distance from a given point called the center. The radius is the distance from the center to any point on the outside of A the circle. The diameter of a circle is the C Q B distance across a circle. AE is a diameter D E A chord is a line segment with endpoints on the circle CD is a chord A diameter is also a chord. QB is a radius

Angles of a Circle <ABC is an inscribed angle. An inscribed angle has its vertex on the circle and with sides that are chords of the circle. AC is an inscribed arc. An inscribed arc is the arc that lies in interior of an inscribed angle and has its endpoints on the angle. <AOB is a central angle. A central angle has its vertex at the center and sides on the circle.

Relationship between Angles and Arcs A central angle has the same measure as its intercepted arc. An inscribed angle is half the measure of its intercepted arc.

Relationship between Angles and Arcs (cont) It follows that an inscribed angle with rays that are on endpoints of the diameter are right angles

Tangent and Secant Lines A tangent line intersects a circle at exactly one point. A secant line intersects a circle at two points What is the difference between a secant line and a chord??

Tangent Lines and Radii The radius of a circle and the tangent line form a right angle at the point the tangent line intersects the circle.

Circumscribed Angles A circumscribed angle is an angle that is formed by two tangent lines. The tangent lines and the radii form a quadrilateral. <J and <S are right angles because they are at the intersection of the tangent line and radii. Because the sum of the angles of a quadrilateral is 360 degrees m<JOS + m<SRJ = 180 A circumscribed angle and its corresponding central angle are supplementary.

Circumscribed Angles (cont) The measure of a circumscribed angle is half the difference of the two arc measures it intersects.

Angles formed by Chords When two chords intersect "inside" a circle, four angles are formed.  At the point of intersection, two sets of vertical angles can be seen in the corners of the X that is formed on the picture.  Remember:  vertical angles are equal. The measure of x is equal to half the sum of the intercepted arcs. X = ½ (mAC + mBD)

Inscribed Quadrilaterals Measure of arcWXY = 138 The rest of the circle is arc WZY. The entire circle is 360 therefore mWZY = 360 – 138 = 222 The inscribed angle of arc WZY is <WXY. m<WXY = ½ (222) = 111 111 + 69 = 180 A Cyclic Quadrilateral's opposite angles add to 180°: a + c = 180° b + d = 180°