Question

your turn

1. find vw
2. find yz

Explanation:

Identify midpoints and midsegments

Based on the tick marks in the diagram:

• Point \(T\) is the midpoint of \(XZ\) because \(XT = TZ\).
• Point \(V\) is the midpoint of \(XY\) because \(XV = VY\).
• Point \(W\) is the midpoint of \(YZ\) because \(YW = WZ\).

Therefore, segment \(VW\) is a midsegment of \(\triangle XYZ\) parallel to side \(XZ\).

Apply the Triangle Midsegment Theorem

The Triangle Midsegment Theorem states that a midsegment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long as that third side:

$$VW = \frac{1}{2}XZ$$

We are given:

$$VW = 4y + 7$$

$$XZ = 12y - 14$$

Substitute these expressions into the midsegment relation:

$$4y + 7 = \frac{1}{2}(12y - 14)$$

Solve for y

Using the Solving Linear Equations knowledge point:

$$
LATEXBLOCK0
$$

Calculate VW

Substitute \(y = 7\) back into the expression for \(VW\):

$$VW = 4(7) + 7 = 35$$

Solve for YZ

Similarly, \(VT\) is a midsegment parallel to \(YZ\), so:

$$VT = \frac{1}{2}YZ$$

We are given:

$$VT = 7x - 17$$

$$YZ = 8x + 2$$

Using the Solving Linear Equations knowledge point:

$$
LATEXBLOCK1
$$

Substitute \(x = 6\) back into the expression for \(YZ\):

$$YZ = 8(6) + 2 = 50$$

Answer:

Question 3

\(VW = 35\)

Question 4

\(YZ = 50\)