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The Distance Formula

The Distance Formula. Used to find the distance between two points: A( x 1 , y 1 ) and B(x 2 , y 2 ) You also could just plot the points and use the Pythagorean Theorem!!. Find the distance between the two points. Round your answer to the nearest tenth. . T(5, 2) and R(-4, -1)

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The Distance Formula

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  1. The Distance Formula • Used to find the distance between two points: A( x1, y1) and B(x2, y2) You also could just plot the points and use the Pythagorean Theorem!!

  2. Find the distance between the two points. Round your answer to the nearest tenth. • T(5, 2) and R(-4, -1) Take a look at example 2, p. 44 2. A( -2, -3) and B(1, 3)

  3. Midpoint Formula • Find the midpoint coordinates between 2 points • Find by averaging the x-coordinates and the y-coordinates of the endpoints (x2, y2) (x1, y1)

  4. Find the coordinates of the midpoint of • Q(3, 5) and S(7, -9) • Q( -4, 4) and S(5, -1)

  5. Special Quadrilaterals Parallelogram – A quadrilateral with both pairs of opposite sides parallel. Rhombus – A parallelogram with four congruent sides. Rectangle – A parallelogram with four right angles. Square – A parallelogram with four congruent sides and four right angles. Kite – A quadrilateral with two pairs of adjacent sides congruent and no opposite sides congruent. Trapezoid – A quadrilateral with exactly one pair of parallel sides. Isosceles Trapezoid – A trapezoid whose nonparallel opposite sides are congruent.

  6. Warm-up Draw each figure on graph paper if possible. If not possible explain why. A parallelogram that is neither a rhombus nor a rectangle An isosceles trapezoid with vertical congruent sides A trapezoid with only one right angle A trapezoid with two right angles A rhombus that is not a square A kite with two right angles

  7. Properties of Parallelograms Theorem 6-1 Opposite sides of a parallelogram are congruent. Theorem 6-2 Opposite angles of a parallelogram are congruent. (Consecutive angles of a parallelogram are supplementary, they are same-side interior angles!) Theorem 6-3 The diagonals of a parallelogram bisect each other.

  8. Using Algebra • Find the value of x in PQRS. Then find QR and PS. 3x - 15 R Q S P 2x + 3

  9. Using Algebra • Find the value of y in EFGH. Then find m<E, m<F, m<G, m<H. E F 3y + 37 6y + 4 G H

  10. TR=12 find VR QS=10 find VS

  11. Find x and the length of the side Find all angle measures 110̇ 120 30

  12. Theorem 6-4 • Theorem 6-4: If three (or more) parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.

  13. Measurement • In the figure, DH ll CG ll BF ll AE, and AB = BC = CD = 2, and EF = 2.5. Find EH. H D 2 G C 2 F B 2 2.5 A E

  14. Proving that a quadrilateral is a parallelogram. (both pairs of opposite sides are parallel) Theorem 6-5 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Theorem 6-6 If one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a parallelogram. Theorem 6-7 If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem 6-8 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

  15. Proving that a quadrilateral is a parallelogram. (Both pairs of opposite sides are parallel) Theorem 6-5 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Theorem 6-6 If one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a parallelogram. Theorem 6-7 If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem 6-8 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

  16. 5 ways to prove that a quadrilateral is a parallelogram. 1. Show that both pairs of opposite sides are || . [definition] 2. Show that both pairs of opposite sides are  . 3. Show that one pair of opposite sides are both  and || . 4. Show that both pairs of opposite angles are  . 5. Show that the diagonals bisect each other .

  17. Examples …… Example 1: Find the value of x and y that ensures the quadrilateral is a parallelogram. y+2 6x = 4x+8 2x = 8 x = 4 units 2y = y+2 y = 2 unit 6x 4x+8 2y Find the value of x and y that ensure the quadrilateral is a parallelogram. Example 2: 5y + 120 = 180 5y = 60 y = 12 units 2x + 8 = 120 2x = 112 x = 56 units (2x + 8)° 120° 5y°

  18. Is the Quadrilateral a Parallelogram? 95° 95° 95 x x

  19. Finding Values • Find the values of a and c for which PQRS must be a parallelogram. Q R a (a + 40) 3c – 3 c + 1 S P

  20. Objective 2: Using Coordinate Geometry • When a figure is in the coordinate plane, you can use the Distance Formula (see—it never goes away) to prove that sides are congruent and you can use the slope formula (see how you use this again?) to prove sides are parallel.

  21. Show that A(2, -1), B(1, 3), C(6, 5) and D(7,1) are the vertices of a parallelogram. Ex. 4: Using properties of parallelograms

  22. Method 1—Show that opposite sides have the same slope, so they are parallel. Slope of AB. 3-(-1) = - 4 1 - 2 Slope of CD. 1 – 5 = - 4 7 – 6 Slope of BC. 5 – 3 = 2 6 - 1 5 Slope of DA. - 1 – 1 = 2 2 - 7 5 AB and CD have the same slope, so they are parallel. Similarly, BC ║ DA. Ex. 4: Using properties of parallelograms Because opposite sides are parallel, ABCD is a parallelogram.

  23. Method 2—Show that opposite sides have the same length. AB=√(1 – 2)2 + [3 – (- 1)2] = √17 CD=√(7 – 6)2 + (1 - 5)2 = √17 BC=√(6 – 1)2 + (5 - 3)2 = √29 DA= √(2 – 7)2 + (-1 - 1)2 = √29 AB ≅ CD and BC ≅ DA. Because both pairs of opposites sides are congruent, ABCD is a parallelogram. Ex. 4: Using properties of parallelograms

  24. Method 3—Show that one pair of opposite sides is congruent and parallel. Slope of AB = Slope of CD = -4 AB=CD = √17 AB and CD are congruent and parallel, so ABCD is a parallelogram. Ex. 4: Using properties of parallelograms

  25. Theorem 6-9 Each diagonal of a rhombus bisects two angles of the rhombus. Theorem 6-10 The diagonals of a rhombus are perpendicular. Theorem 6-11 The diagonals of a rectangle are congruent. Theorem 6-12 If one diagonal of a parallelogram bisects two angles of the parallelogram, then the parallelogram is a rhombus. Theorem 6-13 If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. Theorem 6-14 If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

  26. Use the properties of rectangles to find all missing angle measures, list the properties you used. 32

  27. 6.5: Trapezoids and Kites Objective: To verify and use properties of trapezoids and kites

  28. Trapezoid Midsegment Theorem The midsegment of a trapezoid is parallel to the bases. The length of the midsegment of a trapezoid is half the sum of the lengths of the bases. If you are given the midsegment length and one base: -double the length of the midsegment -subtract the other base.

  29. Midsegment (Median) of a Trapezoid • Joins the midpoints of the nonparallel sides • Is parallel to the bases • Its length is ½ the sum of the bases MN || BC MN || AD MN = ½(BC+AD)

  30. Find the following: EF: 86° 3 4 108°

  31. Find x: x 8 12

  32. Theorem 6-15 The base angles of an isosceles trapezoid are congruent. Theorem 6-16 The diagonals of an isosceles trapezoid are congruent. Theorem 6-17 The diagonals of a kite are perpendicular.

  33. Theorem: The diagonals of a kite are perpendicular. A kite has exactly one pair of opposite, congruent angles.

  34. Find the measure of the missing angles. 1 44° 112° 2 What is the sum of the angles in a quadrilateral?

  35. TRAPEZOID • The 2 parallel sides are the bases • The 2 non-parallel sides are the legs BASE ANGLES B D LEG LEG A C BASE ANGLES Name the following: Bases: Legs: 2 Pairs of Base Angles:

  36. Theorem: The base angles of an isosceles trapezoid are congruent B D C A Find x.

  37. Theorem: The diagonals of an isosceles trapezoid are congruent. EXAMPLE: If BD= 2x+10 and AC=x+15, find x and the length of the diagonals.

  38. 2 angles that share a leg are supplementary because they are same-side interior angles.

  39. The measure of angle A= 110. find the measures of the other 3 angles.

  40. One side of a kite is 4 cm less than two times the length of another side. The perimeter of the kite is 58 cm. Find the lengths of the sides of the kite.

  41. Midsegments of trapezoids • The midsegment of an isosceles trapezoid measures 14 cm. One of the bases measures 24 cm. Find the length of the other base.

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