Lesson-4-proves-the-midline-theorem.pptx

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1. Grade 9 Third Quarter MIDLINE THEOREM 1 2 3 4 5 Lesson 4 Proves the Midline Theorem (M9GEIIId-1)

2. OBJECTIVES 1 2 3 4 5 Illustrates the midline or midsegment of a triangle and trapezoid Applies the Midline Theorem Show accuracy in solving the missing part of a triangle and a trapezoid Grade 9 Third Quarter Lesson 4

3. Activate Prior Knowledge 1 Direction: Given that ABCD is a parallelogram, tell which kind of special parallelogram is identified in the following: SQUARE RHOMBUS RECTANGLE / SQUARE RHOMBUS / SQUARE RECTANGLE / SQUARE ANSWERS KEY

4. Acquire New Knowledge 2 THE MIDLINE THEOREM OF A TRIANGLE In a triangle, a midline (or a midsegment) is any of the three lines joining the midpoints of any pair of the sides of the triangle.

5. Acquire New Knowledge 2 Illustration of the midlines that can be drawn in a triangle. Note: The red segments are the midlines C B A C B A C A B Construction of a midline in a triangle Find the midpoints of the two side Connect the midpoints. Done! The Midline Theorem in a Triangle suggests that the segment that joints the midpoints of two sides of triangle is parallel to the third side and half as long .

6. Acquire New Knowledge 2 C A B The midline is PARALLEL to the third side Illustration: Symbolism: D E > > DE || AC Explanation: This means the midline DE is parallel to AC. Note: Parallel line are marked with “ feathers ” . The symbol “ || ” reads as “is parallel to”

7. Acquire New Knowledge 2 C A B The midline is HALF AS LONG to the third side Illustration: Symbolism: D E 5 cm DE = 1/2 AC or 2DE = AC Explanation: This means the midline is always half as long as the third side (the parallel side to it). In the same manner, the third side is twice as long as the midline 10 cm

8. Acquire New Knowledge 2 Application of Theorem Midline Theorem will always work, on any triangle. It can use when looking other problems. - Some Important Mathematical Ideas

9. Acquire New Knowledge 2 ILLUSTRATIVE EXAMPLE In , O and I are the midpoints of CV and VD, respectively. Consider each given information and answer the question that follow.   V I D C O

10. Acquire New Knowledge 2 ILLUSTRATIVE EXAMPLE V I D C O 1. If OI = 18 CD = _____ How did you solve for CD?_________ OI = CD   18 = CD   2(18) = CD 36 = CD CD = 36 Since OI is the midline, CD should be twice as long. 18 multiplied by 2 is 36. therefore, CD = 36 18 ?

11. Acquire New Knowledge 2 ILLUSTRATIVE EXAMPLE V I D C O 2. If OC = 22.5 VC = _____ How did you solve for VC?_________ OC + OV = CV Since O is the midpoint of VC, therefore OC and OV are equal and by adding these two segments, VC measures 45. ? 22.5 22.5 + 22.5 = CV 45= CV CV = 45

12. Acquire New Knowledge 2 ILLUSTRATIVE EXAMPLE V I D C O 3. If CD = 34 OI = _____ How did you solve for OI?_________ Since OI is the midline, it has to be half of CD. 34 divided by 2 is equal to 17. therefore, OI = 13 ? 34 OI = CD   OI = (34)   OI = 17

13. Activate Prior Knowledge 2 Pass and Sing Every rectangle is a square. FALSE 2. Every square is a rectangle. TRUE 3. Every square is a rhombus. TRUE 4. Every rhombus is a square. FALSE 5. The diagonals of a rhombus are perpendicular to each other. TRUE  

14. Activate Prior Knowledge 2 Pass and Sing 6 . Every parallelogram is a square. FALSE 7. Every square is a parallelogram. TRUE 8. A rhombus is equiangular. FALSE 9. If the diagonals of a given parallelogram are perpendicular, then the parallelogram is a rectangle. FALSE 10. A square is an equiangular and equilateral parallelogram. TRUE  

15. Acquire New Knowledge 2 Another kind of quadrilateral that is equally important as parallelogram is the trapezoid and kite. A trapezoid is quadrilateral with exactly one pair of parallel sides . The parallel side of a trapezoid are called the bases and the non-parallel sides are called legs . The angle formed by a base and a leg are called base angle . You have to prove some theorems on trapezoids. In a trapezoid, a midline ( midsegment or median) is a line that joins the midpoint of the sides that are not parallel THE MIDLINE THEOREM OF A TRAPEZOID

16. Acquire New Knowledge 2 Illustration of the midline that can be drawn in a trapezoid. Note: The red segments are the midlines. The midline theorem in a trapezoid suggest that the segment connecting the midpoints of the two legs of a trapezoid is parallel to the bases , and its length is equal to half the sum of lengths of the bases .

17. Acquire New Knowledge 2 B The midline is PARALLEL to the bases Illustration: Symbolism: Explanation: This means that the midline ED is parallel to the bases AB and DC. A F C D E AB || EF , DC || EF

18. Acquire New Knowledge 2 Midline Theorem will always work, on any trapezoid. It can use when looking other problems. Application of Theorem - Some Important Mathematical Ideas

19. Acquire New Knowledge 2 ILLUSTRATIVE EXAMPLE: In Trapezoid LOVE, Y and U are midpoints of LE and OV, respectively. Consider each given information and answer the questions that follow. E V U O L Y

20. Acquire New Knowledge 2 ILLUSTRATIVE EXAMPLE 1. If LO = 8 and VE = 6 YU = __ How did you solve for YU?______ Since YU is the midline, YU should be half of the sum of the bases. Half of the sum of 8 and 6 is 7. Therefore YU = 7 YU =   E V U O L Y ? 8 YU =   YU =   YU = 7 6

21. Acquire New Knowledge 2 ILLUSTRATIVE EXAMPLE 2. If YU = 10 and LO = 13 VE = _____ How did you solve for VE?_______ Since YU is the midline, VE should be obtain using the same formula. Substitute the given, multiplied both sides by 2, transposed 13 and the result is 7. Therefore, YU = 7 YU =   E V U O L Y 10 13 10 =   20 = 13 + VE 20 - 13 = VE ? 7 = VE VE = 7

22. Acquire New Knowledge 2 ILLUSTRATIVE EXAMPLE: L M Q N O P If │PQ│= 20 cm, │LM│ = x+3, and │ON│ = x+6, what is the value of x?

23. Acquire New Knowledge 2 ILLUSTRATIVE EXAMPLE: L M Q N O P If │LM│= 2x+2 cm, │PQ│=3x+3, and │ON│ = 2(x+6), what is │LM│?

24. Application 3 a 48 24 6 12 b c d Direction : Choose the correct answer. E D 1. In ∆SIE, L and D are the midpoints of SI and IE respectively. If LD is 12, then what is the value of SE? I S L

25. Application 3 a 28 21 7 14 b c d Direction : Choose the correct answer. E D 2. In ∆SIE, L and D are the midpoints of SI and IE respectively. If SE = 14, then what is the value of LD? I S L

26. Application 3 a OU = BS   b d e Direction : Choose the correct answer. S U 3. In ∆BNS, O and U are the midpoints of BN and NS respectively. How will you compute the OU? N B O OU = US   OU = BN   OU = ON  

27. Application 3 a BO≅ON b c d Direction : Choose the correct answer. S U 4. In ∆BNS, O and U are the midpoints of BN and NS respectively. List the congruent sides. N B O BO≅US ON≅UN NU≅US

28. 4 a AE ≅ DE EF ≅ AB BF ≅ FC AD ≅ BC b c d Direction : Choose the correct answer. 5. In figure below, E and F are mid-points of AD and BC respectively. List the congruent sides. A B C D F E Application

29. Assessment 4 a DE IL LS SE b d e Direction : Choose the correct answer. E D 1. In figure below, D is mid-point of IE and LD|| SE then ID is equal to I S L

30. Assessment 4 a 8.2 cm 5.1 cm 4.1 cm 4.9 cm b c d Direction : Choose the correct answer. 2. In figure below, D and E are mid-points of AB and AC respectively. The length of DE is: A B C D E 4.9 cm 5.1 cm 8.2 cm

31. 3 a OU | | UN b c d Direction : Choose the correct answer. S U 3. In ∆BNS, O and U are the midpoints of BN and NS respectively. List the parallel sides. N B O BO | | US ON | | UN OU | | BS Assessment

32. Assessment 4 a 18 cm 8 cm 9 cm 19 cm b c d Direction : Choose the correct answer. 4. In figure below, E and F are mid-points of AD and BC respectively. The length of EF is: A B C D F 7 cm 11 cm E 4 cm

33. Assessment 4 a 18 cm 8 cm 9 cm 19 cm b c d Direction : Choose the correct answer. 5. In figure below, E and F are mid-points of AD and BC respectively. The length of DA is: A B C D F 7 cm 11 cm E 4 cm

34. Additional Activity 5 In trapezoid ABCD, EF is the midsegment. Apply your knowledge about midsegment of a trapezoid by writing a check mark corresponding to the statement whether it is NOT A TRAP for true statement, and TRAP for false statement A E D B F C 22 cm 48 cm 70 cm 62 cm

35. Additional Activity 5 A E D B F C 22 cm 48 cm 70 cm 62 cm STATEMENT NOT A TRAP TRAP 1) EF = 118 CM  2) EF || AB || DC  ANSWERS KEY

36. Additional Activity 5 A E D B F C 22 cm 48 cm 70 cm 62 cm STATEMENT NOT A TRAP TRAP 3) AE = 44 CM  4) FB = 31 CM  5) DC EF  STATEMENT NOT A TRAP TRAP 3) AE = 44 CM  4) FB = 31 CM  

37. References OTHER SUPPORTING LEARNING MATERIALS: Compendium of notes Instructional Support Materials In Mathematics INTERNET: https://quizizz.com/admin/quiz/5df41634198380001bea369d/the-midline-theorem http://aven.amritalearning.com/index.php?sub=102&brch=305&sim=1593&cnt=3871