Math Problem Statement

CDE is an isosceles triangle with D  E . If CD = 4x + 9, DE = 7x – 5, and CE = 16x – 27, find x and the measure of each side.

Solution

In an isosceles triangle, two sides are equal in length. Since CDE\triangle CDE is isosceles with DE\angle D \cong \angle E, we know that sides CDCD and CECE are equal.

Thus, we set up the equation: CD=CECD = CE Substitute the given expressions for CDCD and CECE: 4x+9=16x274x + 9 = 16x - 27

Step 1: Solve for xx

  1. Rearrange the equation to isolate xx: 9+27=16x4x9 + 27 = 16x - 4x 36=12x36 = 12x
  2. Divide by 12: x=3x = 3

Step 2: Find the Measure of Each Side

Now that we know x=3x = 3, substitute it into each expression for CDCD, DEDE, and CECE to find their lengths.

  1. For CDCD: CD=4x+9=4(3)+9=12+9=21CD = 4x + 9 = 4(3) + 9 = 12 + 9 = 21

  2. For DEDE: DE=7x5=7(3)5=215=16DE = 7x - 5 = 7(3) - 5 = 21 - 5 = 16

  3. For CECE: CE=16x27=16(3)27=4827=21CE = 16x - 27 = 16(3) - 27 = 48 - 27 = 21

Solution Summary

  • x=3x = 3
  • CD=21CD = 21
  • DE=16DE = 16
  • CE=21CE = 21

The measures of the sides are:

  • CD=21CD = 21
  • DE=16DE = 16
  • CE=21CE = 21

Would you like more details on any step or have further questions?

Here are five related questions to deepen your understanding:

  1. What properties define an isosceles triangle, and why do they apply here?
  2. How would the solution change if it were specified that DE=CDDE = CD instead?
  3. What are the implications of equal angles in an isosceles triangle?
  4. Can you find the perimeter of CDE\triangle CDE with the values found?
  5. How does solving for xx help determine all unknowns in such geometry problems?

Tip: Always check your final values by substituting them back into original equations to verify consistency.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Algebra
Isosceles Triangles
Equations

Formulas

Equation for equal sides in isosceles triangle
Linear equation solution

Theorems

Isosceles triangle properties

Suitable Grade Level

Grades 7-9