Question

1. △cde is an isosceles triangle with \\( \angle d \cong \angle e \\). if \\( cd = 4x + 9, de = 7x - 5 \\), and \\( ce = 16x - 27 \\), find \\( x \\) and the measure of each side. \\( x = \\)___ \\( cd = \\)_ \\( de = \\)_ \\( ce = \\)___

Explanation:

Step1: Identify the equal sides

Since \(\angle D\cong\angle E\), then \(CD = CE\) (In an isosceles triangle, sides opposite congruent angles are congruent).
So, \(4x + 9=16x-27\)

Step2: Solve for \(x\)

Subtract \(4x\) from both sides:
\(4x + 9-4x=16x-27 - 4x\)
\(9 = 12x-27\)
Add \(27\) to both sides:
\(9 + 27=12x-27+27\)
\(36 = 12x\)
Divide both sides by \(12\):
\(x=\frac{36}{12}=3\)

Step3: Find the length of \(CD\)

Substitute \(x = 3\) into \(CD=4x + 9\)
\(CD=4\times3+9=12 + 9=21\)

Step4: Find the length of \(DE\)

Substitute \(x = 3\) into \(DE=7x-5\)
\(DE=7\times3-5=21 - 5=16\)

Step5: Find the length of \(CE\)

Substitute \(x = 3\) into \(CE=16x-27\)
\(CE=16\times3-27=48 - 27=21\)

Answer:

\(x = 3\)
\(CD=21\)
\(DE = 16\)
\(CE=21\)