Math Problem Statement

In the figure, ZY ≅ XW, ZX = 5x + 17, YW = 10 - 2x, and YX = 3. Find ZY and XW.

Solution

Let's analyze the given information from the problem.

  • ZYXW\overline{ZY} \cong \overline{XW}, which means the lengths of segments ZYZY and XWXW are equal.
  • ZX=5x+17ZX = 5x + 17
  • YW=102xYW = 10 - 2x
  • YX=3YX = 3

We are asked to find the lengths of ZYZY and XWXW, knowing that ZY=XWZY = XW.

Step-by-step approach:

  1. Since ZYXWZY \cong XW, we know that ZY=XWZY = XW. The length of ZYZY can be expressed as the difference between ZXZX and XYXY. So: ZY=ZXXY=(5x+17)3=5x+14ZY = ZX - XY = (5x + 17) - 3 = 5x + 14
  2. Similarly, the length of XWXW can be expressed as the difference between YWYW and XYXY: XW=YWXY=(102x)3=72xXW = YW - XY = (10 - 2x) - 3 = 7 - 2x
  3. Since ZY=XWZY = XW, we can set the two equations equal to each other: 5x+14=72x5x + 14 = 7 - 2x
  4. Solve for xx: 5x+2x=7145x + 2x = 7 - 14 7x=77x = -7 x=1x = -1
  5. Now substitute x=1x = -1 back into the expressions for ZYZY and XWXW to find their lengths: ZY=5(1)+14=5+14=9ZY = 5(-1) + 14 = -5 + 14 = 9 XW=72(1)=7+2=9XW = 7 - 2(-1) = 7 + 2 = 9

Therefore, both ZYZY and XWXW have lengths of 9 units.

Final Answer:

ZY=XW=9ZY = XW = 9

Would you like a further explanation or details on any step?

Here are five related questions to deepen your understanding:

  1. How would the problem change if ZXZX was a quadratic function of xx?
  2. Can you find the value of YWYW using the value of xx?
  3. What would happen if ZY\overline{ZY} and XW\overline{XW} were not congruent?
  4. How does the equality of segment lengths ZYXW\overline{ZY} \cong \overline{XW} help solve this problem?
  5. How would you graph the relationships between these segments on a coordinate plane?

Tip: Always double-check if geometric figures suggest congruence or equality in length; it can simplify solving equations in geometry problems.

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Math Problem Analysis

Mathematical Concepts

Geometry
Congruence
Algebraic Equations

Formulas

ZY = ZX - YX
XW = YW - YX
ZY = XW

Theorems

Congruent Segments Theorem

Suitable Grade Level

Grades 9-10