Math Problem Statement

EF is a midsegment of isosceles trapezoid ABCD. Find each measure.

Solution

To solve the given problem systematically, we need to analyze the information provided in the isosceles trapezoid ABCD and apply the properties of a trapezoid and its midsegment.


Given Information

  1. AB=8x43AB = 8x - 43 (Top base)
  2. CD=10CD = 10 (Bottom base)
  3. EF=2xEF = 2x (Midsegment)
  4. AE=2y+3AE = 2y + 3 (Left side segment)
  5. BF=6y1BF = 6y - 1 (Right side segment)
  6. EFEF is a midsegment.

Key Properties

  1. Midsegment Rule for a trapezoid: EF=AB+CD2EF = \frac{AB + CD}{2} where EFEF is the midsegment, ABAB and CDCD are the bases.

  2. Since the trapezoid is isosceles, the non-parallel sides are equal: AE=BFAE = BF


Step 1: Solve for xx

Using the midsegment formula: EF=AB+CD2EF = \frac{AB + CD}{2} Substitute the given expressions: 2x=(8x43)+1022x = \frac{(8x - 43) + 10}{2}

Multiply through by 2 to eliminate the denominator: 4x=(8x43)+104x = (8x - 43) + 10 Simplify: 4x=8x334x = 8x - 33

Solve for xx: 4x8x=334x - 8x = -33 4x=33x=334orx=8.25-4x = -33 \quad \Rightarrow \quad x = \frac{33}{4} \quad \text{or} \quad x = 8.25


Step 2: Solve for yy

Since AE=BFAE = BF, substitute the given expressions: 2y+3=6y12y + 3 = 6y - 1

Solve for yy: 2y6y=132y - 6y = -1 - 3 4y=4y=1-4y = -4 \quad \Rightarrow \quad y = 1


Step 3: Find Each Measure

Now substitute x=8.25x = 8.25 and y=1y = 1 into the expressions:

  1. AEAE: AE=2y+3=2(1)+3=5AE = 2y + 3 = 2(1) + 3 = 5

  2. EDED: Since EE and FF are midpoints of ADAD and BCBC, ED=AE=5ED = AE = 5.

  3. BFBF: BF=6y1=6(1)1=5BF = 6y - 1 = 6(1) - 1 = 5

  4. FCFC: By symmetry, FC=BF=5FC = BF = 5.

  5. ABAB: AB=8x43=8(8.25)43AB = 8x - 43 = 8(8.25) - 43 Simplify: AB=6643=23AB = 66 - 43 = 23

  6. EFEF: EF=2x=2(8.25)=16.5EF = 2x = 2(8.25) = 16.5


Final Answers

  1. x=8.25x = 8.25
  2. y=1y = 1
  3. AE=5AE = 5
  4. ED=5ED = 5
  5. BF=5BF = 5
  6. FC=5FC = 5
  7. AB=23AB = 23
  8. EF=16.5EF = 16.5

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Related Questions

  1. How do we derive the midsegment formula for a trapezoid?
  2. What defines an isosceles trapezoid, and how does symmetry help solve problems?
  3. What are some real-world applications of midsegments in geometry?
  4. How can systems of equations help solve geometric problems?
  5. How does the congruence of sides AEAE and BFBF simplify the solution?

Tip: Always use symmetry in isosceles trapezoid problems to reduce the number of unknowns.

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Math Problem Analysis

Mathematical Concepts

Isosceles Trapezoid
Midsegment in Geometry
Algebraic Equations

Formulas

Midsegment of a trapezoid: EF = (AB + CD) / 2
Equating non-parallel sides in an isosceles trapezoid: AE = BF

Theorems

Midsegment Theorem for Trapezoids
Congruence of Legs in Isosceles Trapezoids

Suitable Grade Level

Grades 8-10