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Three Theorems Involving Proportions

Three Theorems Involving Proportions. Section 8.5. F. Prove similar triangles first. Given: AT ║ BN FA = 12, FT = 15 AT = 14, AB = 8 Find: length BN & NT. A. T. B. N. =. BN = 23 1 / 3 TN = 10. =.

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Three Theorems Involving Proportions

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  1. Three Theorems Involving Proportions Section 8.5

  2. F Prove similar triangles first Given: AT ║ BN FA = 12, FT = 15 AT = 14, AB = 8 Find: length BN & NT A T B N = • BN = 23 1/3 • TN = 10 =

  3. Theorem 65: If a line is ║to one side of a Δ and intersects the other 2 sides, it divides those 2 sides proportionally. (Side splitter theorem)

  4. Given: AB CD EF Conclusion:     = A B D C G E F Draw AF In Δ EAF, = In Δ ABF, = Substitute BD for AG and DF for GF, substitution or transitive =

  5. Theorem 66: If three or more parallel lines are intersected by two transversals, the parallel lines divide the transversals proportionally.

  6. = A Given: ∆ ABD AC bisects <BAD Prove: D B C = Set up Proportions = BC • AD = AB • CD Means Extreme Therefore: = =

  7. Theorem 67: If a ray bisects an angle of a triangle, it divides the opposite side into segments that are proportional to the adjacent sides. (Angle Bisector Theorem)

  8. a P W Given: a, b, c and d are parallel lines, Lengths given, WZ = 15 Find: WX, XY and YZ 2 I X b 3 G Y c 4 d S Z According to Theorem 66, the ratio of WX:XY:YZ is 2:3:4. Therefore, let WX = 2x, XY = 3x and YZ = 4x 2x + 3x + 4x = 15 9x = 15 x = 5/3 WX = 10/3 = 3 1/3 XY = 5 YZ = 20/3 = 6 2/3

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