Question

−x + 25
3(2x − 8)
a
b
c
60°
30°
d

Explanation:

1. First, note the property of the perpendicular - bisector:

• Since \(BD\) is the perpendicular - bisector of \(AC\), \(\triangle ADC\) is an isosceles triangle with \(AD = CD\), and \(\angle A=\angle C = 60^{\circ}\), so \(\triangle ADC\) is an equilateral triangle. Also, \(AB = BC\).
• We know that the angles in \(\triangle ABC\) are related to the given expressions. Since \(\angle A=\angle C = 60^{\circ}\), we can set up an equation based on the fact that the angles in a triangle sum to \(180^{\circ}\). But we can also use the property of the equal - length segments formed by the perpendicular - bisector.
• Since \(AB = BC\), we can set up the equation \(-x + 25=3(2x - 8)\).

1. Expand the right - hand side of the equation:

• Expand \(3(2x - 8)\) using the distributive property \(a(b - c)=ab - ac\). Here, \(a = 3\), \(b = 2x\), and \(c = 8\), so \(3(2x - 8)=6x-24\).
• Our equation becomes \(-x + 25=6x-24\).

1. Add \(x\) to both sides of the equation:

• \(-x+x + 25=6x+x-24\), which simplifies to \(25 = 7x-24\).

1. Add 24 to both sides of the equation:

• \(25 + 24=7x-24 + 24\), so \(49 = 7x\).

1. Solve for \(x\):

• Divide both sides of the equation \(49 = 7x\) by 7. We get \(x=\frac{49}{7}=7\).

Step1: Set up the equation

Since \(AB = BC\), we have \(-x + 25=3(2x - 8)\).

Step2: Expand the right - hand side

Using the distributive property, \(3(2x - 8)=6x - 24\), so the equation is \(-x + 25=6x-24\).

Step3: Add \(x\) to both sides

To get all \(x\) terms on one side, \(-x+x + 25=6x+x-24\), resulting in \(25 = 7x-24\).

Step4: Add 24 to both sides

To isolate the term with \(x\), \(25 + 24=7x-24 + 24\), giving \(49 = 7x\).

Step5: Solve for \(x\)

Divide both sides by 7, \(x=\frac{49}{7}=7\).

Answer:

\(x = 7\)