10.1 Tangents to Circles Geometry Objectives/Assignment • Recall the basic properties of circle. • Identify segments and lines related to circles. • Use properties of a tangent to a circle. • Assignment: Worksheet and Problems from Exercise not done in class. Some definitions you need • 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. Some definitions you need • The distance across the circle, through its center is the diameter of the circle. The diameter is twice the radius. • The terms radius and diameter describe segments as well as measures. center diameter radius Some definitions you need • A radius is a segment whose endpoints are the center of the circle and a point on the circle. • QP, QR, and QS are radii of Q. All radii of a circle are congruent. S P Q R Some definitions you need • 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. S P Q R 1.Secant of the circle • A line which intersects a circle at two distinct points is called a secant of the circle. B A What is the difference between a chord and a secant of the circle ? B B A A 2.Tangent of the circle • A line which intersects(meets) the circle at exactly one point is called a tangent of the circle. A Ex. 1: Identifying Special Segments and Lines Tell whether the line or segment is best described as a chord, a secant, a tangent, a diameter, or a radius of the circle. a. AD b. CD c. EG d. HB K B A J C D E H F G 3.Concentric circles • Circles that have a common center are called concentric circles. No points of intersection Concentric circles Using properties of tangents 4.The point at which a tangent line intersects(meets or touches) the circle to which it is tangent is called the point of contact or point of tangency. 5.Thus, the point of contact is the common point of the tangent to the circle and the circle. 6.Theorem 10.1 • The tangent at any point of a circle is perpendicular to the radius through the point of contact. • If l is tangent to the circle with centre Q at point P, then l ⊥QP. P Q l 7.Conversely, • 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 the circle. P Q l 8.Also, • Since there can be only one perpendicular to QP at the point P, it follows that one and only one tangent can be drawn to a circle at a given point on the circumference. • In fact to draw a tangent to the circle at P, join QP and draw a line perpendicular to QP at P. P Q l 9. And • Since there can be only one perpendicular to l at the point P, it follows that the perpendicular to a tangent at its point of contact passes through the centre. P Q l 10. Prove that the tangents drawn at the ends of a diameter of a circle are parallel. • Given: • To prove: • Proof: l 11. Prove that the segment joining the points of contact of two parallel tangents of a circle, passes through the centre. • Given: • To prove: • Proof: Ex. 4: Verifying a Tangent to a Circle • In the figure, D is the centre of the circle. E is a point on the circle. F is a point outside the circle such that DF = 61 units and EF = 60 units. Is EF a tangent to the circle? D 61 11 E 60 F 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? B 16 ft. r C 8 ft. A r B 16 ft. Solution: r A c2 = a2 + b2 (r + 8)2 = r2 + 162 r 2 + 16r + 64 = r2 + 256 16r + 64 = 256 16r = 192 r = 12 C 8 ft. r Pythagorean Thm. Substitute values Square of binomial Subtract r2 from each side. Subtract 64 from each side. Divide. The radius of the silo is 12 feet. Prove that in two concentric circles, the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact • Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle. NUMBER OF TANGENTS TO A CIRCLE PASSING THROUGH A POINT (say P) Case 1: P is inside the circle The number of tangents possible is …………… Case 2: P is on the circle • Recall point no. 8 • Since there can be only one perpendicular to QP at the point P, it follows that one and only one tangent can be drawn to a circle at a given point on the circumference. P Q Case 3. P is outside the circle • From a point in the circle’s exterior, you can draw exactly two different tangents to the circle. • Recall : Angle in a semicircle is a right angle. • Or the angle subtended by a diameter on the circumference is a right angle. Length of a tangent from a point outside the circle • It is the distance between the point and the point of contact. In the figure below PT is the length of the tangent. T P Theorem 10.3: The lengths of tangents drawn from an external point to a circle are equal. • Chord of contact • The line segment joining the points of contact of two tangents drawn from a point outside the circle is called a chord of contact. • In the figure above, prove the following: 1. OP bisects angle APB. 2. OAPB is a cyclic quadrilateral. 3. Angle APB and angle AOB are supplementary. 4. Angle APB = 2 (angle BAO). 5. OP bisects AB at right angles. 6. If AB = 8 cm and OA = 5 cm, find PA. Circle inscribed in a triangle or A triangle circumscribing a circle. • Let r be the radius of the circle and a, b, c be the sides of the triangle. s abc 2 • Show that area of triangle = r X s. • s= BD + CE + AF • A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6 cm respectively. Find the sides AB and AC. Circle inscribed in a quadrilateral or A quadrilateral circumscribing a circle. • Show that AB + CD = BC + AD • Parallelogram subscribing a circle is a rhombus. • Opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle. Textbook problem no. 9 • In Fig., XY and X′Y′ are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X′Y′ at B. Prove that ∠ AOB = 90°.