 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.

In an isosceles triangle, two sides are equal in length. Since \( \triangle CDE \) is isosceles with \( \angle D \cong \angle E \), we know that sides \( CD \) and \( CE \) are equal.

Thus, we set up the equation:
\[
CD = CE
\]
Substitute the given expressions for \( CD \) and \( CE \):
\[
4x + 9 = 16x - 27
\]

### Step 1: Solve for \( x \)
1. Rearrange the equation to isolate \( x \):
   \[
   9 + 27 = 16x - 4x
   \]
   \[
   36 = 12x
   \]
2. Divide by 12:
   \[
   x = 3
   \]

### Step 2: Find the Measure of Each Side
Now that we know \( x = 3 \), substitute it into each expression for \( CD \), \( DE \), and \( CE \) to find their lengths.

1. **For \( CD \):**
   \[
   CD = 4x + 9 = 4(3) + 9 = 12 + 9 = 21
   \]

2. **For \( DE \):**
   \[
   DE = 7x - 5 = 7(3) - 5 = 21 - 5 = 16
   \]

3. **For \( CE \):**
   \[
   CE = 16x - 27 = 16(3) - 27 = 48 - 27 = 21
   \]

### Solution Summary
- \( x = 3 \)
- \( CD = 21 \)
- \( DE = 16 \)
- \( CE = 21 \)

The measures of the sides are:
- \( CD = 21 \)
- \( DE = 16 \)
- \( CE = 21 \)

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 \( DE = CD \) instead?
3. What are the implications of equal angles in an isosceles triangle?
4. Can you find the perimeter of \( \triangle CDE \) with the values found?
5. How does solving for \( x \) help determine all unknowns in such geometry problems?

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

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.

Learn how to solve for x in an isosceles triangle where two sides are equal. This step-by-step guide shows how to find the lengths of each side, given the equations for CD, DE, and CE. The solution also highlights key algebraic concepts and triangle properties. Suitable for students in Grades 7-9.

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