theorems on tangents, Secants and segments of a circles 1.pptx

Theorems on tangents and secants

1. MATHEMATICS 10 Sir Pee Jay

2. E B A C I am a line that intersects a circle in exactly one point WHAT AM I? Tangent Line

3. I am a line that intersects a circle in exactly two point. WHAT AM I? K Y A B X Secant Line

4. I am a part of secant segment that is outside the circle. WHAT AM I? A B M C External Secant Segment

5. I am a region bounded by an arc of circle and two radii to the endpoints of Arc WHAT AM I ? B C A Sector of A Circle

6. Direction: This game is called word search and all you have to do is search a word that is related to mathematics and give some insight about the word you search WARM-UP Exercise

7. Objectives At the end of the lesson, the students can: Understand the theorems on secants, tangents and segments of a circle; Value Accumulated knowledge as means of new understanding; Solve and proves problems involving secant segment, tangent segment and external secant segment theorems Solve and proves theorems on angle formed by secants and tangents

8. THEOREMS on s ecants, tangents and segments of a circle

9. THEOREMS is a true statement that can be proven . POSTULATES is a statement that is assumed true without proof.

10. If X is a given point on the circle, there is only a single line which can be drawn through X that is tangent to the circle. Postulate on Tangents

11. Theorems on Tangents OR is a tangent line and point U is the point of tangency. If OR is tangent to Circle T at point U, then it is perpendicular to Radius TU.     1

12. Theorems on Tangents 2

13. Theorems on Tangents 3

14. C A B D If AC = 10cm, then what is the length of BC ? Solution: AC = 10cm AC ≅ BC therefore BC = 10cm If two tangents segments is drawn from the point outside the circle, then the segments are congruent

15. Theorems on angles formed by tangents and secants

16. The measure of the angle formed by two secants that intersects outside the circle is one-half the positive difference of the two intercepted arcs 1

17. The measure of the angle formed by two secants that intersects outside the circle is one-half the positive difference of the two intercepted arcs 1. If m FC = 96° and m EB = 32°, what is m ∠FDC? ⌒ ⌒ m ∠FDC=? m ∠FDC= ½ ( mFC – mEB ) ⌒ ⌒ m ∠FDC= ½ (96° – 32° ) m ∠FDC= ½ (64°) m ∠FDC= 32°

18. The measure of the angle formed by a secant and a tangent that intersect outside the circle is one-half the positive difference of the two intercepted arcs. 2

19. The measure of the angle formed by a secant and a tangent that intersect outside the circle is one-half the positive difference of the two intercepted arcs. 1. If mDFC = 220° and mDB = 80°, what is m ∠DEC? ⌒ ⌒ m ∠DEC=? m ∠DEC= ½ ( mDFC – mDB ) ⌒ ⌒ m ∠DEC= ½ (220° – 80°) m ∠DEC= ½ (140°) m ∠DEC= 70°

20. The measure of angle formed by two tangents that intersects outside the circle is one-half the positive difference of two intercepted arcs. Figure 7 3 In Figure 7 at the right, EP and DP are two tangents that intersects outside the circle at point P, EFD and ED are the two intercepted arcs of ∠EPD m∠EPD = ½( mEFD – mED ) If mEPD = 214 and mED = 46 °, then m ∠EPD =? m∠EPD = ½(214 ° - 46°) m∠EPD = ½(168 °) m ∠EPD = 84 ° °

21. The measure of angle formed by two tangents that intersects outside the circle is one-half the positive difference of two intercepted arcs. 1. If mHOD = 216° and mHD = 66°, what is m ∠HFD? ⌒ ⌒ m ∠HFD=? m ∠HFD= ½ ( mHOD – mHD ) ⌒ ⌒ m ∠HFD= ½ (216° – 66°) m ∠HFD= ½ (150°) m ∠HFD= 75°

22. The measure angle formed by two secants that intersects inside the circle is one-half the sum of the measures of the two intercepted arcs and its vertical angle In Figure 8 at the right, EC and PY are two secants that intersects inside the circle at point A, EY and PC are the two intercepted arcs of ∠EAY and ∠PAC. EP and YC are the two intercepted arcs of ∠EAP and ∠YAC m∠EAY = ½ ( mEY+mPC ) if mEY =92 ° and mPC = 196 , What is m ∠EAY and ∠YAC? m∠EAY = ½ ( mEY+mPC ) m∠EAY = ½ (92 °+196°) m∠EAY = ½ (288 °) m∠EAY = 144 ° ° 4 Figure 8 m∠YAC =? if two angles formed a linear pair, the angles are supplementary m∠EAY + m∠YAC = 180 ° 144° + m∠YAC = 180 ° m∠YAC = 180 ° - 144 ° m∠YAC =36 °

23. The measure angle formed by two secants that intersects inside the circle is one-half the sum of the measures of the two intercepted arcs and its vertical angle 1 . If mEB = 45° and mCD = 49°, what is m ∠EFB? m ∠BFD? ⌒ ⌒ m ∠EFB=? m ∠EFB= ½ ( mEB + mCD ) ⌒ ⌒ m ∠EFB= ½ (94°) m ∠EFB= 47° m ∠EFB= ½ (45° + 49°) m∠BFD =? if two angles formed a linear pair, the angles are supplementary m∠EFB + m∠BFD = 180 ° 47° + m∠BFD = 180 ° m∠BFD = 180 ° - 47 ° m∠BFD =133 °

24. The measure of the angle formed by a secant and tangent that intersect at the point of tangency is half the measure of its intercepted arc. In Figure 9 at the right, IA is a tangent and GH is a secant intersect at point G which is the point of tangency. GOH is the intercepted arc of ∠IGH m∠IGH = ½( mGOH ) If mGOH = 232 , what is the m∠IGH ? m∠IGH = ½ ( mGOH ) m∠IGH = ½ (232 °) m∠IGH = 116 ° ° Figure 9 5

25. The measure of the angle formed by a secant and tangent that intersect at the point of tangency is half the measure of its intercepted arc. 1 . If mBFD = 216° , what is m ∠DBE? m ∠DBE=? m ∠DBE= ½ ( mBDF ) ⌒ ⌒ m ∠DBE= ½ (216° ) m ∠DBE= 108°

26. Theorems on Secant Segment, tangent segment and external secant Segment

27. If two secant segments are drawn to a circle from the same exterior point, then the product of the lengths is of one secant segment and its external secant segment is equal to the product of the lengths of the other secant segment and its external secant segment. In Figure 10 at the right, AE and CE are a secant segment drawn from exterior point E. Therefore, AE ● BE = CE ● DE. If the lengths of AE=10, BE=4 and CE= 8 DE=x, What is the length of DE? AE ● BE = CE ● DE 10 ● 4 = 8 ● x 40 = 8x 5 = x 1 Figure 10   Therefore DE = 5

28. If two secant segments are drawn to a circle from the same exterior point, then the product of the lengths is of one secant segment and its external secant segment is equal to the product of the lengths of the other secant segment and its external secant segment. 1 . If the lengths of DC=16, EC=5 and BC= 10 FC=x , What is the length of FC? DC ● EC = BC ● FC Therefore the length of FC = 8 16 ● 5 = 10 ● x 80 = 10x =   8 = x

29. If tangent segment and secant segment are drawn to a circle from the same exterior point, then the square of the length of the tangent segment is equal to the product of the lengths of the secant segment and its external segment . In Figure 11 at the right, ML is a tangent segment and KL is a secant segment drawn from the same exterior point which is point L. Therefore = KL ● NL If KL = 9 and NL = 5, Find ML = KL ● NL = 9 ● 5 = ML = ML = 3   2 Figure 11

30. If tangent segment and secant segment are drawn to a circle from the same exterior point, then the square of the length of the tangent segment is equal to the product of the lengths of the secant segment and its external segment. If the lengths of BD=6, CD=9 and ED= x, What is the length of ED? = CD ● ED   Therefore the length of ED = 4 = 9 ● x   36 = 9x =   4 = x

31. Formative Assessment

32. C A B D If BC = 15cm, then what is the length of AC ? Solution: B C = 15cm BC ≅ A C therefore AC = 15cm If two tangents segments is drawn from the point outside the circle, then the segments are congruent

33. The measure of the angle formed by two secants that intersects outside the circle is one-half the positive difference of the two intercepted arcs If m DB = 80° and mEF = 30°, what is m ∠DCB? ⌒ ⌒ m ∠DCB=? m ∠DCB= ½ ( mDB – mEF ) ⌒ ⌒ m ∠DCB= ½ (80° – 30° ) m ∠DDB= ½ (50°) m ∠DCB= 25°

34. The measure of the angle formed by a secant and a tangent that intersect outside the circle is one-half the positive difference of the two intercepted arcs. If mHOD = 160° and mHE = 50°, what is m ∠HLD? ⌒ ⌒ m ∠HLD=? m ∠HLD= ½ ( mHOD – mHE ) ⌒ ⌒ m ∠HLD= ½ (160° – 50°) m ∠HLD= ½ (110°) m ∠HLD= 55 °

35. The measure of the angle formed by a secant and a tangent that intersect outside the circle is one-half the positive difference of the two intercepted arcs. If mDFC = 200° and mDB = 50°, what is m ∠DEC? ⌒ ⌒ m ∠DEC=? m ∠DEC= ½ ( mDFC – mDB ) ⌒ ⌒ m ∠DEC= ½ (200° – 50°) m ∠DEC= ½ (150°) m ∠DEC= 75°

36. The measure of angle formed by two tangents that intersects outside the circle is one-half the positive difference of two intercepted arcs. If mCEB = 200° and mCB = 54°, what is m ∠CDB? ⌒ ⌒ m ∠CDB=? m ∠CDB= ½ ( mCEB – mCB ) ⌒ ⌒ m ∠CDB= ½ (200° – 54°) m ∠CDB= ½ (146°) m ∠CDB= 73°