Question

1. $\triangle cde$ is an isosceles triangle with $\overline{cd} \cong \overline{de}$. if $cd = 9x - 25$, $de = 6x + 8$, and $ce = 10x - 73$, find the value of $x$ and the measure of each side.

Explanation:

Step1: Set CD equal to DE (since CD ≅ DE)

Since \( \overline{CD} \cong \overline{DE} \), their lengths are equal. So we set up the equation:
\( 9x - 25 = 6x + 8 \)

Step2: Solve for x

Subtract \( 6x \) from both sides:
\( 9x - 6x - 25 = 8 \)
\( 3x - 25 = 8 \)

Add 25 to both sides:
\( 3x = 8 + 25 \)
\( 3x = 33 \)

Divide both sides by 3:
\( x = \frac{33}{3} \)
\( x = 11 \)

Step3: Find the length of CD

Substitute \( x = 11 \) into \( CD = 9x - 25 \):
\( CD = 9(11) - 25 \)
\( CD = 99 - 25 \)
\( CD = 74 \)

Step4: Find the length of DE

Since \( CD \cong DE \), \( DE = CD = 74 \). (Or substitute \( x = 11 \) into \( DE = 6x + 8 \):
\( DE = 6(11) + 8 \)
\( DE = 66 + 8 \)
\( DE = 74 \))

Step5: Find the length of CE

Substitute \( x = 11 \) into \( CE = 10x - 73 \):
\( CE = 10(11) - 73 \)
\( CE = 110 - 73 \)
\( CE = 37 \)

Answer:

The value of \( x \) is \( 11 \). The lengths of the sides are \( CD = 74 \), \( DE = 74 \), and \( CE = 37 \).