![]() We can also have more than one secant to the circle interacting with each other, like this: We can see that the line intersects the outside of the circle at two points and creates a line segment between those points, which is highlighted in red. A line cutting across a circle that touches two points on the outside of the circle is a secant to the circle. Let’s start by defining the vocabulary we’ll need. Therefore, ON = 3.Hello! Today we’re going to explore what can happen when a circle and a line or two lines meet. OM can now be found by the use of the Pythagorean Theorem or by recognizing a Pythagorean triple. Therefore, m = 27 ½ and m = 152 ½°.Įxample 5: Use Figure 8, in which AB = 8, CD = 8, and OA = 5, to find ON.įigure 8 A circle with two chords equal in measure.īy Theorem 81, ON = OM. Since m ∠ AOB = 55°, that would make m = 55° and m = 305°. Since OA = 13 and AM = 5, OM can be found by using the Pythagorean Theorem.Īlso, Theorem 80 says that m = m and m = m. įigure 7 A circle with a diameter perpendicular to a chord. In Figure 5, if OX = OY, then by Theorem 82, AB = CD.įigure 5 A circle with two minor arcs equal in measure.Įxample 3: Use Figure 6, in which m = 115°, m = 115°, and BD = 10, to find AC.įigure 6 A circle with two minor arcs equal in measure.Įxample 4: Use Figure 7, in which AB = 10, OA = 13, and m ∠ AOB = 55°, to find OM, m and m. Theorem 82: In a circle, if two chords are equidistant from the center of a circle, then the two chords are equal in measure. In Figure 4, if AB = CD, then by Theorem 81, OX = OY.įigure 4 In a circle, the relationship between two chords being equal in measure and being equidistant from the center. Theorem 81: In a circle, if two chords are equal in measure, then they are equidistant from the center. įigure 3 A diameter that is perpendicular to a chord. In Figure 3, UT, diameter QS is perpendicular to chord QS By Theorem 80, QR = RS, m = m, and m = m. Theorem 80: If a diameter is perpendicular to a chord, then it bisects the chord and its arcs. These theorems can be used to solve many types of problems. Some additional theorems about chords in a circle are presented below without explanation. (b) If m = and EF = 8, find GH.įigure 2 The relationship between equality of the measures of (nondiameter) chords and equality of the measures of their corresponding minor arcs. Theorem 79: In a circle, if two minor arcs are equal in measure, then their corresponding chords are equal in measure.Įxample 1: Use Figure 2 to determine the following. The converse of this theorem is also true. ![]() Theorem 78: In a circle, if two chords are equal in measure, then their corresponding minor arcs are equal in measure. This is stated as a theorem.įigure 1 A circle with four radii and two chords drawn. This would make m ∠1 = m ∠2, which in turn would make m = m. In Figure 1, circle O has radii OA, OB, OC and OD If chords AB and CD are of equal length, it can be shown that Δ AOB ≅ Δ DOC. Summary of Coordinate Geometry Formulas.Slopes: Parallel and Perpendicular Lines.Similar Triangles: Perimeters and Areas. ![]() Proportional Parts of Similar Triangles. ![]()
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