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.
2. $\triangle wxy$ is an isosceles triangle with $\overline{wx} \cong \overline{wy}$. if $wx$ is 3 more than four times $x$, $xy$ is 7 less than five times $x$, and $wy$ is 66 less than seven times $x$, find $x$ and the measure of each side.

Explanation:

Problem 14:

Step1: Set CD = DE (isosceles triangle)

Since \( \triangle CDE \) is isosceles with \( \overline{CD} \cong \overline{DE} \), we set their expressions equal:
\( 9x - 25 = 6x + 8 \)

Step2: Solve for x

Subtract \( 6x \) from both sides:
\( 3x - 25 = 8 \)
Add 25 to both sides:
\( 3x = 33 \)
Divide by 3:
\( x = 11 \)

Step3: Find side lengths

• \( CD = 9(11) - 25 = 99 - 25 = 74 \)
• \( DE = 6(11) + 8 = 66 + 8 = 74 \) (matches CD)
• \( CE = 10(11) - 73 = 110 - 73 = 37 \)

Problem 15:

Step1: Define side lengths

• \( WX = 4x + 3 \) (3 more than 4x)
• \( WY = WX = 4x + 3 \) (isosceles, \( \overline{WX} \cong \overline{WY} \))
• \( XY = 5x - 7 \) (7 less than 5x) Wait, correction: "XY is 7 less than five times x" → \( XY = 5x - 7 \)? Wait, original: "XY is 7 less than five times x" → \( XY = 5x - 7 \)? Wait, re-express:

Wait, original: "WX is 3 more than four times x" → \( WX = 4x + 3 \)
"XY is 7 less than five times x" → \( XY = 5x - 7 \)
"WY is 66 less than seven times x" → \( WY = 7x - 66 \)
But \( WX = WY \), so:

Step2: Set WX = WY

\( 4x + 3 = 7x - 66 \)

Step3: Solve for x

Subtract \( 4x \):
\( 3 = 3x - 66 \)
Add 66:
\( 69 = 3x \)
Divide by 3:
\( x = 23 \)

Step4: Find side lengths

• \( WX = 4(23) + 3 = 92 + 3 = 95 \)
• \( WY = 7(23) - 66 = 161 - 66 = 95 \) (matches WX)
• \( XY = 5(23) - 7 = 115 - 7 = 108 \)

Answer:

s:

Problem 14:

• \( x = 11 \)
• \( CD = 74 \), \( DE = 74 \), \( CE = 37 \)

Problem 15:

• \( x = 23 \)
• \( WX = 95 \), \( WY = 95 \), \( XY = 108 \)