Question

in the triangles, \\(tr = ge\\) and \\(sr = fe\\).
if \\(\overline{gf} = 3.2\text{ ft}\\), which is a possible measure of \\(\overline{ts}\\)?

• \\(1.6\text{ ft}\\)
• \\(3.0\text{ ft}\\)
• \\(3.2\text{ ft}\\)
• \\(4.0\text{ ft}\\)

Explanation:

Identify given geometric relationships

We are given two triangles, \(\triangle TSR\) and \(\triangle GFE\), with:

• \(TR = GE\) (indicated by single tick marks)
• \(SR = FE\) (indicated by double tick marks)
• \(m\angle R = 56^\circ\)
• \(m\angle E = 42^\circ\)
• \(GF = 3.2\text{ ft}\)

Apply the Hinge Theorem

The Hinge Theorem (SAS Inequality Theorem) states that if two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first triangle is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle.

• The included angle for sides \(TR\) and \(SR\) is \(\angle R\).
• The included angle for sides \(GE\) and \(FE\) is \(\angle E\).
• Since \(56^\circ > 42^\circ\), we have \(m\angle R > m\angle E\).
• Therefore, the opposite side \(TS\) must be strictly greater than \(GF\).

Set up the inequality

Using the Hinge Theorem:

$$TS > GF$$

Given \(GF = 3.2\text{ ft}\):

$$TS > 3.2\text{ ft}$$

Evaluate the given options

We compare the options to the inequality \(TS > 3.2\text{ ft}\):

• \(1.6\text{ ft}\) (not greater than \(3.2\))
• \(3.0\text{ ft}\) (not greater than \(3.2\))
• \(3.2\text{ ft}\) (not strictly greater than \(3.2\))
• \(4.0\text{ ft}\) (strictly greater than \(3.2\))

Thus, \(4.0\text{ ft}\) is the only possible measure.

Answer:

• (A) 1.6 ft
• (B) 3.0 ft
• (C) 3.2 ft
• (D) 4.0 ft (Correct answer)