Question

which of the following statements must be true? select all that apply.
a bd bisects ac
b △abc is isosceles.
c bd is the perpendicular bisector of ac
d ad = 12

Explanation:

Step1: Set up equation for equal - length segments

Since \(AD = 4x\) and \(DC=x + 9\), and \(AD=DC\) (from the double - hash marks indicating equal lengths), we set up the equation \(4x=x + 9\).
\[4x=x + 9\]

Step2: Solve the equation for \(x\)

Subtract \(x\) from both sides: \(4x−x=x + 9−x\), which gives \(3x=9\). Then divide both sides by 3: \(x = 3\).
\[3x=9\Rightarrow x = 3\]

Step3: Analyze each statement

• Statement A: Since \(AD = DC\) (because \(4x=x + 9\) and we can solve for \(x\) to show they are equal), by the definition of a bisector (a line that divides a segment into two equal parts), \(\overline{BD}\) bisects \(\overline{AC}\). This statement is True.
• Statement B: The tick - marks on \(AB\) and \(BC\) indicate that \(AB = BC\). By the definition of an isosceles triangle (a triangle with at least two equal - length sides), \(\triangle ABC\) is isosceles. This statement is True.
• Statement C: We know that \(AD = DC\) (from the equal - length markings) and \(\angle BDA=\angle BDC = 90^{\circ}\) (the right - angle symbol). By the definition of a perpendicular bisector (a line that is perpendicular to a segment and divides it into two equal parts), \(\overline{BD}\) is the perpendicular bisector of \(\overline{AC}\). This statement is True.
• Statement D: If \(x = 3\), then \(AD=4x=4\times3 = 12\). This statement is True.

Answer:

A. \(\overline{BD}\) bisects \(\overline{AC}\), B. \(\triangle ABC\) is isosceles, C. \(\overline{BD}\) is the perpendicular bisector of \(\overline{AC}\), D. \(AD = 12\)