# Solve problems involving tangents to a circle. Solve problems involving chords of a circle

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1 8UNIT ircle Geometry What You ll Learn How to Solve problems involving tangents to a circle Solve problems involving chords of a circle Solve problems involving the measures of angles in a circle Why Is It Important? ircle properties are used by artists, when they create designs and logos Key Words radius (radii) right angle tangent point of tangency diameter right triangle isosceles triangle chord perpendicular bisector central angle inscribed angle arc subtended semicircle 37

2 8. Skill uilder Solving for Unknown Measures in Triangles Here are ways to find unknown measures in triangles. ngle Sum roperty ythagorean Theorem In any triangle: In any right QR: b q r c a R p a b c 80 q p r Q Here is how to find the unknown measures in right QR. In QR, the angles add up to 80. To find, start at 80 and subtract the known measures y the ythagorean Theorem: QR R Q 8 q 7 So: q 8 7 q So, is 30 and q is about 4 cm. q R 60 8 cm 7 cm nswer to the same degree of accuracy as the question uses. Q heck. Find each unknown measure. a) b) 3.0 cm 5.0 cm x x x So, x is. 38

3 8. roperties of Tangents to a ircle FUS Use the relationship between tangents and radii to solve problems. tangent touches a circle at exactly one point. Tangent Radius oint of tangency Tangent-Radius roperty tangent to a circle is perpendicular to the radius drawn to the point of tangency. So,, 90 and 90 means perpendicular to. Example Finding the Measure of an ngle in a Triangle is tangent to the circle at. is the centre of the circle. Find the measure of. 50 Solution y the tangent-radius property: 90 Since the sum of the angles in is 80 : So, is 40. heck. Find the value of D 39

4 Example Using the ythagorean Theorem in a ircle M is a tangent to the circle at. is the centre. Find the length of radius. r 0 cm 8 cm M Solution y the tangent-radius property: M 90 y the ythagorean Theorem in right M: M M 0 r 8 00 r r 36 r 36 r r 6 Radius has length 6 cm. heck. ST is a tangent to the circle at S. is the centre. Find the length of radius S. nswer to the nearest millimetre. r S cm 6 cm T ST T y the tangent-radius property y the ythagorean Theorem r r r r r r S is about cm long. 30

5 ractice In each question, is the centre of the circle.. From the diagram, identify: J H E F G D a) 3 radii,, b) tangents, c) points of tangency, d) 4 right angles,,,. What is the measure of each angle? a) b) Q R Q R 3. Find each value of. a) b) W 35 T T 5 M TW 80 3

6 4. Find each value of x. nswer to the nearest tenth of a unit. a) x 0 km km T T 90 x x x x y the tangent-radius property y the ythagorean Theorem in T x x So, is about km. b) c) 4 cm 6 cm x Q 5 cm x 5 cm M Q, and: x x x x x So, Q is about cm. x x x x x So, is about cm. 3

7 8. roperties of hords in a ircle FUS Use chords and related radii to solve problems. chord of a circle joins points on the circle. Radius hord Diameter hord roperties In any circle with centre and chord : If bisects, then. If, then. The perpendicular bisector of goes through the centre. Example Finding the Measure of ngles in a Triangle Find,, and z. 30 Solution z bisects chord, so Therefore, 90 y the angle sum property in : Since radii are equal,, and is isosceles. So, z z In an isosceles triangle, base angles are equal. 33

8 heck. Find the values of and. 40 So, y the chord properties y the angle sum property. Find the values of,, and z. 55 z So, y the chord properties y the angle sum property Since, is isosceles and So, z 34

9 Example Using the ythagorean Theorem in a ircle is the centre of the circle. Find the length of chord. 6 cm 0 cm Solution 6 cm 0 cm 0 6 y the ythagorean Theorem in right So, 8 cm Since, bisects. y the chord properties So, 8 cm The length of chord is: 8 cm 6 cm 35

10 heck. Find the values of a and b. E 5 cm b G 3 cm a F a y the ythagorean Theorem in right FG So, a cm y the chord properties So, b cm ractice In each diagram, is the centre of the circle.. Name all radii, chords, and diameters. a) D b) R T S Q N Radii: hords: Diameters: Radii: hords: Diameters: 36

11 . n each diagram, mark line segments with equal lengths. Then find each value of a. a) b) 4 cm a M a 3 cm N cm So, a cm MN cm cm So, a c) K d) a 8 cm T.5 cm a S M L L S cm cm So, a cm So, a 37

12 3. Find each value of and. a) b) Q Q is 80 So, Find the length of chord. 5. Find N. 5 cm cm D 7 cm N 30 cm D y the ythagorean Theorem D So, D cm cm y the chord properties So, chord has length: cm cm N cm y the chord properties cm N y the ythagorean Theorem N So, N is cm. 38

13 HEKINT an you Solve problems using tangent properties? Solve problems using chord properties? 8. In each diagram, is the centre of the circle. ssume that lines that appear to be tangent are tangent.. Name the angles that measure 90. a) b) G M E M N. Find the unknown angle measures. a) b) 75 s M p q N p Tangent-radius property q 80 ngle sum property q 90 s s 39

14 3. Find the values of a and b to the nearest tenth. a) b) Q 5 cm a 0 cm 5 cm b 0 cm Q y the tangent-radius property Q is of Q. a So, a cm y the ythagorean Theorem is of. b So, b cm Find the unknown measures. a) X b) b M 6 cm d L N 35 Y X XY is. So, b MN MN cm cm So, d cm 330

15 5. Find each value of,, and z. z 65 y the chord properties y the angle sum property, so is isosceles. So, z 6. Find the length of. 0 cm Q 6 cm R Q QR y the chord properties cm cm Q y the ythagorean Theorem So, the length of is cm. 33

16 8.3 roperties of ngles in a ircle FUS In a circle: Use inscribed angles and central angles to solve problems. Inscribed angle central angle has its vertex at the centre. n inscribed angle has its vertex on the circle. entral angle oth angles in the diagram are subtended by arc. rc entral ngle and Inscribed ngle roperty The measure of a central angle is twice the measure of an inscribed angle subtended by the same arc. So,, or Inscribed ngles roperty Inscribed angles subtended by the same arc are equal. So, D E D E 33

17 Example Using Inscribed and entral ngles Find the values of and. D Solution D entral and inscribed are both subtended by arc. So, 44 and D are inscribed angles subtended by the same arc. So, D 333

18 heck. Find each value of. a) b) c) D D. Find the values of and. a) T b) S 30 D Q 70 Q QS QT D 334

19 ngles in a Semicircle roperty Inscribed angles subtended by a semicircle H are right angles. F G H 90 F G Example Finding ngles in an Inscribed Triangle Find and. N 40 I M Solution MIN is an inscribed angle subtended by a semicircle. So, y the angle sum property in MIN 50 heck. Find the values of and

20 ractice. Name the following from the diagram. a) the central angle subtended by arc : b) the central angle and inscribed angle subtended by arc D: and c) the inscribed angle subtended by a semicircle: d) the right angle: D. In each circle, name a central angle and an inscribed angle subtended by the same arc. Shade the arc. a) b) Q R entral angle: Inscribed angle: entral angle: Inscribed angle: 3. Determine each indicated measure. a) G F b) 84 S 8 T R H GF GHF TSR c) E d) 7 D F L J G K DEG 336

21 4. Determine each value of and. a) 34 b) 5 5. Find the value of and y the angle sum property 6. Find the value of,, and z. z 50 In, is. In : z y the angle sum property So, and z 337

22 Unit 8 uzzle ircle Geometry Word Search R E S E M I I R L E E K L T S E S Y E L N S L I N F Y E H L G I M U U U R E E R S N G N F M Q G T R R I T N T I E L N D E L S E E K E D S E V F F E D L L T N N E T R E T M I S E Z D E L E T E L U T T I S E L N R N Z M D V U J E D R K E G G E E G U R M M E R E Z T E E L G N I R T N M E R D I U S I G H Q R T N I F R G R Q Y E L G N D E I R S N I Find these words in the puzzle above. You can move in any direction to find the entire word. letter may be used in more than one word. degrees radius perpendicular centre inscribed angle bisect diameter central angle circle tangent chord triangle circumference point angle arc isosceles equal point of tangency semicircle subtended 338

23 Unit 8 Study Guide Skill Description Example Recognize and apply tangent properties Recognize and apply chord properties in circles 6 cm 53 y K 0 5 M 8 cm L 30 Recognize and apply angle properties in a circle If, then. If, then. Inscribed and central angles, or Inscribed angles 90 and 60 ML 0 5 y 50 z z 00 D E D E ngles on a semicircle D E D E

24 Unit 8 Review 8.. Find each value of and. Segments RS and MN are tangents. a) R b) 0 S M 75 N NM Find each value of x to the nearest tenth. Segments GH and ST are tangents. a) H b) S x 3 cm 4 cm cm 5 cm G x T HG x ST So, x cm So, x cm 340

25 8. 3. Find the values of and. y the chord properties y the angle sum property 5 4. Find the values of,, and z. y the M N, so is isosceles. M 7 z N N M So, z z 5. Find the length of the radius of the circle to the nearest tenth. XY cm Y cm X Z cm cm Draw radius X. X XY X X The radius is about cm. 34

26 Find each value of. a) b) c) Find each value of and. a) b) Find the value of w,,, and z. w 5 z y the angle sum property z D z D is. So, D D w w w y the angle sum in D w w w 34

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