Question

1. (overline{de}) is a perpendicular bisector. write and solve an equation to find (x). then find (ce) and (cd).
2. (py) is an angle bisector. write and solve an equation to find (x). then find (cx), (xw), and (cw).

Explanation:

Problem 3

Step1: Use the property of perpendicular bisector

Since \( \overline{AE} \) is a perpendicular bisector, \( CE = DE \). So \( 5x-15=3x + 1 \).

Step2: Solve the equation for \( x \)

Subtract \( 3x \) from both sides: \( 5x-3x-15=3x - 3x+1 \), \( 2x-15 = 1 \). Add 15 to both sides: \( 2x=1 + 15 \), \( 2x=16 \). Divide by 2: \( x=\frac{16}{2}=8 \).

Step3: Find \( CE \) and \( DE \)

Substitute \( x = 8 \) into \( CE=5x-15 \): \( CE=5\times8-15=40 - 15=25 \). Substitute \( x = 8 \) into \( DE=3x + 1 \): \( DE=3\times8+1=24 + 1=25 \).

Step1: Use the angle - bisector theorem

By the angle - bisector theorem, \( \frac{GX}{XW}=\frac{GU}{UW} \). Let \( GX=XW \) (since \( \overline{FX} \) is an angle bisector and \( \triangle GXU\) and \( \triangle WXU\) have some congruent properties related to the angle bisector and the side - angle - side (SAS) if we consider the angle bisected and the common side \( \overline{XU}\)), but more directly from the angle - bisector theorem \( \frac{GX}{XW}=\frac{GU}{UW}\). Since \( GU = 3x+8\) and \( UW = 80\), and \( GX = XW\) (because of the angle - bisector and the congruent triangles formed by the angle bisector and the perpendiculars from a point on the bisector to the sides of the angle), we can also set up the equation \( 3x+8=80\).

Step2: Solve the equation for \( x \)

Subtract 8 from both sides: \( 3x=80 - 8\), \( 3x=72\). Divide by 3: \( x = 24\).

Step3: Find \( GX\), \(XW\) and \(GW\)

\( GX=3x + 8\), substitute \( x = 24\): \( GX=3\times24+8=72 + 8=80\). Since \( GX = XW\) (angle - bisector property for the lengths of the segments adjacent to the angle being bisected when considering the ratio from the angle - bisector theorem and the fact that the triangles formed are congruent), \( XW = 80\). \( GW=GX+XW=80 + 80=160\).

Answer:

\( x = 8\), \(CE = 25\), \(DE = 25\)

Problem 4