Question

name all the points shown that must be midpoints of the sides of the large triangle.
7.
8.
9.

Explanation:

Step1: Analyze Diagram 7

In triangle 7, segments \(AD = DB = x\) and \(AE = EC = y\). By the definition of midpoint (a point that divides a segment into two equal parts), \(D\) is the midpoint of \(AB\) and \(E\) is the midpoint of \(AC\). For \(F\), the base segments are \(z\) and \(z + 1\), which are not equal, so \(F\) is not a midpoint.

Step2: Analyze Diagram 8

In triangle 8, \(AD = DB = 9\), so \(D\) is the midpoint of \(AB\). \(AE = 5\), but the other segment on \(AC\) isn't given as equal, but \(DE\) and \(DF\) - Wait, no, \(AD = DB = 9\), so \(D\) is midpoint of \(AB\). \(EF\) and \(DF\) - Wait, actually, \(AD = DB = 9\), so \(D\) is midpoint of \(AB\). \(E\): The side \(AC\) has \(AE = 5\), but the other part? Wait, no, in the diagram, \(AD = DB = 9\), so \(D\) is midpoint of \(AB\). \(F\): The base has \(6\) and then \(F\) - Wait, no, \(AD = DB = 9\), so \(D\) is midpoint of \(AB\). \(E\): Wait, maybe I misread. Wait, in diagram 8, \(AD = DB = 9\), so \(D\) is midpoint of \(AB\). \(E\): Is \(AE = EC\)? No, but \(D\) is midpoint of \(AB\) because \(AD = DB = 9\).

Step3: Analyze Diagram 9

In triangle 9, the base has equal segments (marked with ticks) at \(F\), so \(F\) is midpoint of the base. \(D\) and \(E\): The segments on the other sides are marked with ticks, so \(D\) is midpoint of one side, \(E\) midpoint of another, and \(F\) midpoint of the base. Wait, but the question is "name all the points shown that must be midpoints".

Wait, re - evaluating:

For Diagram 7:
- \(D\): \(AD = DB\) (both \(x\)), so \(D\) is midpoint of \(AB\).
- \(E\): \(AE = EC\) (both \(y\)), so \(E\) is midpoint of \(AC\).
- \(F\): \(BF\) and \(FC\) are \(z\) and \(z + 1\), not equal, so not midpoint.

For Diagram 8:
- \(D\): \(AD = DB = 9\), so \(D\) is midpoint of \(AB\).
- \(E\): Not enough info? Wait, no, \(AD = DB = 9\), so \(D\) is midpoint.

For Diagram 9:
- \(D\): The side has equal ticks, so \(D\) is midpoint.
- \(E\): The side has equal ticks, so \(E\) is midpoint.
- \(F\): The base has equal ticks, so \(F\) is midpoint.

But the question is about each diagram (7, 8, 9). Wait, the original problem has three diagrams: 7, 8, 9.

For Diagram 7:
Points \(D\) (midpoint of \(AB\)) and \(E\) (midpoint of \(AC\)) must be midpoints because \(AD = DB\) and \(AE = EC\). \(F\) is not, as \(z eq z + 1\).

For Diagram 8:
Point \(D\) is midpoint of \(AB\) because \(AD = DB = 9\).

For Diagram 9:
Points \(D\), \(E\), and \(F\) are midpoints because the sides have equal - length markings (ticks) indicating they divide the sides into two equal parts.

But the question is "Name all the points shown that must be midpoints of the sides of the large triangle." for each diagram? Wait, the user provided three diagrams: 7, 8, 9.

Wait, let's re - check Diagram 7:

In triangle \(ABC\) (let's name the large triangle \(ABC\)), \(D\) on \(AB\), \(E\) on \(AC\), \(F\) on \(BC\).

\(AD = DB = x\), so \(D\) is midpoint of \(AB\).

\(AE = EC = y\), so \(E\) is midpoint of \(AC\).

\(BF = z\), \(FC = z + 1\), so \(BF eq FC\), so \(F\) is not a midpoint.

Diagram 8:

Triangle \(ABC\), \(D\) on \(AB\), \(E\) on \(AC\), \(F\) on \(BC\).

\(AD = DB = 9\), so \(D\) is midpoint of \(AB\).

Diagram 9:

Triangle \(ABC\), \(D\) on \(AB\), \(E\) on \(AC\), \(F\) on \(BC\).

The markings on \(AB\) (at \(D\)), \(AC\) (at \(E\)), and \(BC\) (at \(F\)) show equal segments, so \(D\) is midpoint of \(AB\), \(E\) midpoint of \(AC\), \(F\) midpoint of \(BC\).

But the question is probably for each diagram. Let's assume the first diagram (7) as the main? Wait, the user's diagram:

Wait, the problem is "Name all the points shown that must be midpoints of the sides of the large triangle."

For diagram 7:

- \(D\) (since \(AD = DB\))
- \(E\) (since \(AE = EC\))

For diagram 8:

- \(D\) (since \(AD = DB = 9\))

For diagram 9:

- \(D\), \(E\), \(F\) (since all sides have equal - length markings at these points)

But maybe the question is about diagram 7 (the first one). Let's focus on diagram 7:

In diagram 7, \(AD = DB\) (both \(x\)) and \(AE = EC\) (both \(y\)). By the definition of a midpoint (a point that divides a segment into two congruent segments), \(D\) is the midpoint of \(\overline{AB}\) and \(E\) is the midpoint of \(\overline{AC}\). The segment \(BF\) has length \(z\) and \(FC\) has length \(z + 1\), so \(F\) is not a midpoint.

Answer:

In diagram 7: \(D\) (midpoint of \(AB\)), \(E\) (midpoint of \(AC\)); In diagram 8: \(D\) (midpoint of \(AB\)); In diagram 9: \(D\), \(E\), \(F\) (midpoints of their respective sides)

(If we consider the first diagram (7) as the main, the answer is \(D\) and \(E\))