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Tangents Chapter 10 Section 5
Recall What is a Circle –set of all points in a plane that are equidistant from a given point called a center of the circle. A circle with center P is called “circle P”, or P. The distance from the center to a point on the circle is called the radius of the circle. Two circles are congruent if they have the same radius.
Remember A chord is a segment whose endpoints are points on the circle. PS and PR are chords. A diameter is a chord that passes through the center of the circle. PR is a diameter.
Tangent A tangent is – a line in the plane of a circle that intersects the circle in exactly one point. –Line l is a tangent.
Ex. 1: Identifying Special Segments and Lines Tell whether the line or segment is best described as a chord, a tangent, a diameter, or a radius of C. a.AD b.CD c.EG d.HB
More information you need-- In a plane, two circles can intersect in two points, one point, or no points. Coplanar circles that intersect in one point are called tangent circles. Coplanar circles that have a common center are called concentric. 2 points of intersection.
Tangent circles A line or segment that is tangent to two coplanar circles is called a common tangent. A common internal tangent intersects the segment that joins the centers of the two circles. A common external tangent does not intersect the segment that joins the center of the two circles. Internally tangent
Concentric circles Circles that have a common center are called concentric circles. Concentric circles No points of intersection
Ex. 2: Identifying common tangents Tell whether the common tangents are internal or external.
Ex. 2: Identifying common tangents Tell whether the common tangents are internal or external. The lines m and n do not intersect AB, so they are common external tangents. In a plane, the interior of a circle consists of the points that are inside the circle. The exterior of a circle consists of the points that are outside the circle.
Ex. 3: Circles in Coordinate Geometry Give the center and the radius of each circle. Describe the intersection of the two circles and describe all common tangents.
Ex. 3: Circles in Coordinate Geometry Center of circle A is (4, 4), and its radius is 4. The center of circle B is (5, 4) and its radius is 3. The two circles have one point of intersection (8, 4). The vertical line x = 8 is the only common tangent of the two circles.
Using properties of tangents The point at which a tangent line intersects the circle to which it is tangent is called the point of tangency.
Theorem 10.91 If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. If l is tangent to Q at point P, then l ⊥ QP. l
Theorem 10.10 In a plane, if a line is perpendicular to a radius of a circle at its endpoint on a circle, then the line is tangent to the circle. If l ⊥ QP at P, then l is tangent to Q. l
Ex. 4: Verifying a Tangent to a Circle You can use the Converse of the Pythagorean Theorem to tell whether EF is tangent to D. Because 11 2 + 60 2 = 61 2, ∆DEF is a right triangle so DE is perpendicular to EF. So by Theorem 10.10; EF is tangent to D.
Ex. 5: Finding the radius of a circle You are standing at C, 8 feet away from a grain silo. The distance from you to a point of tangency is 16 feet. What is the radius of the silo? First draw it. Tangent BC is perpendicular to radius AB at B, so ∆ABC is a right triangle; So you can use the Pythagorean theorem to solve.
Solution: (r + 8) 2 = r 2 + 16 2 Pythagorean Thm. Substitute values c 2 = a 2 + b 2 r 2 + 16r + 64 = r 2 + 256Square of binomial 16r + 64 = 256 16r = 192 r = 12 Subtract r 2 from each side. Subtract 64 from each side. Divide. The radius of the silo is 12 feet.
Note: From a point in the circle’s exterior, you can draw exactly two different tangents to the circle. The following theorem tells you that the segments joining the external point to the two points of tangency are congruent.
Theorem 10.11 If two segments from the same exterior point are tangent to the circle, then they are congruent. IF SR and ST are tangent to P, then SR ST.
Ex. 7: Using properties of tangents AB is tangent to C at B. AD is tangent to C at D. Find the value of x. x 2 + 2
Solution: x 2 + 2 11 = x 2 + 2 Two tangent segments from the same point are Substitute values AB = AD 9 = x 2 Subtract 2 from each side. 3 = xFind the square root of 9. The value of x is 3 or -3.
Construction Construct a line tangent to a circle through an exterior point –http://www.mathopenref.com/consttangent s.htmlhttp://www.mathopenref.com/consttangent s.html Construct a line tangent to a circle through a point on the circle –http://www.mathopenref.com/consttangent. htmlhttp://www.mathopenref.com/consttangent. html
Circumscribed Polygons Remember circles can be circumscribed around polygons Polygons can be circumscribed around circles Instead of the vertices lying on the circle, every side of the polygon is tangent to the circle!
Secants, Tangents and Angle Measures Chapter 10 Section 6
Secants Secant- –A line that intersects a circle in exactly 2 points –Extension of a chord
Theorem 10.13 If a tangent and a secant intersect at a the point of tangency, then the measure of each angle formed is one half the measure of its intercepted arc. m 1= ½m m 2= ½m
Ex. 1: Finding Angle and Arc Measures Line m is tangent to the circle. Find the measure of the red angle or arc. Solution: m 1= ½ m 1= ½ (150 °) m 1= 75 ° 150°
Ex. 1: Finding Angle and Arc Measures Line m is tangent to the circle. Find the measure of the red angle or arc. Solution: m = 2(130 °) m = 260 ° 130°
Ex. 2: Finding an Angle Measure In the diagram below, is tangent to the circle. Find m CBD Solution: m CBD = ½ m 5x = ½(9x + 20) 10x = 9x +20 x = 20 m CBD = 5(20 °) = 100° (9x + 20)° 5x° D
Lines Intersecting Inside or Outside a Circle If two lines intersect a circle, there are three (3) places where the lines can intersect. on the circle
Inside the circle
Outside the circle
Lines Intersecting You know how to find angle and arc measures when lines intersect ON THE CIRCLE. You can use the following theorems to find the measures when the lines intersect INSIDE or OUTSIDE the circle.
Theorem 10.12 If two secants intersect in the interior of a circle, then –The measure of an angle formed is ½ the sum of the measures of the arcs intercepted by the angle and its vertical angle – 2
Theorem 10.14 If a tangent and a secant, two tangents or two secants intercept in the EXTERIOR of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs. m 1 = ½ m( - m )
Theorem 10.14 If a tangent and a secant, two tangents or two secants intercept in the EXTERIOR of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs. m 2 = ½ m( - m )
Theorem 10.14 If a tangent and a secant, two tangents or two secants intercept in the EXTERIOR of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs. m 3 = ½ m( - m ) 3
Ex. 3: Finding the Measure of an Angle Formed by Two Chords Find the value of x Solution: x ° = ½ (m +m x ° = ½ (106° + 174°) x = 140 Apply Theorem 10.12 Substitute values Simplify 174 ° 106 ° x°x°
Ex. 4: Using Theorem 10.14 Find the value of x Solution: 72 ° = ½ (200° - x°) 144 = 200 - x ° - 56 = -x 56 = x Substitute values. Subtract 200 from both sides. Multiply each side by 2. m GHF = ½ m( - m ) Apply Theorem 10.14 Divide by -1 to eliminate negatives. 200 ° x°x° 72 °
Ex. 4: Using Theorem 10.14 Find the value of x Solution: = ½ (268 - 92) = ½ (176) = 88 Substitute values. Multiply Subtract m GHF = ½ m( - m ) Apply Theorem 10.14 x°x° 92 ° Because and make a whole circle, m =360 °-92°=268°
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