Question

1. triangle trs was constructed from the radii of two congruent overlapping circles, circle r and circle s.

tq bisects rs and ∠rts. which of the following statements can not be concluded from the given figure?
options:

• ( mangle stq = mangle rtq = 30^circ )
• ( moverline{tq} = \frac{1}{3}(moverline{tr} + moverline{rs} + moverline{st}) )
• triangle trs is an equilateral triangle.
• ( moverline{rq} = moverline{sq} = \frac{1}{2}moverline{ts} )

Explanation:

Step1: Analyze Triangle TRS

Since Circle R and Circle S are congruent, \( TR = TS = RS \) (radii of congruent circles), so \( \triangle TRS \) is equilateral. This makes the third option valid.

Step2: Analyze Angle Bisector and Perpendicular Bisector

\( TQ \) bisects \( \angle RTS \) and \( \overline{RS} \) (given, and \( TQ \perp RS \) as seen). In an equilateral triangle, the angle bisector, median, and altitude coincide. So \( \angle STQ=\angle RTQ = 30^\circ \) (since \( \angle RTS = 60^\circ \), bisected into two \( 30^\circ \) angles), making the first option valid. Also, \( RQ = SQ=\frac{1}{2}RS \), and since \( RS = TS \) (equilateral), \( RQ = SQ=\frac{1}{2}TS \), so the fourth option is valid.

Step3: Analyze the Arc Measure Formula

For the second option, \( m\widehat{TQ} \) is a straight - line segment? No, \( \widehat{TQ} \) is an arc? Wait, no, \( TQ \) is a segment. Wait, actually, let's check the formula. The sum \( m\widehat{TR}+m\widehat{RS}+m\widehat{ST} \): \( \widehat{TR} \) and \( \widehat{ST} \) are radii - related arcs, and \( \widehat{RS} \) is a chord - related arc? Wait, in an equilateral triangle \( TRS \), \( m\widehat{TR}=m\widehat{ST}=m\widehat{RS} \) (since \( TR = TS = RS \) as radii of congruent circles). Let's assume each of \( m\widehat{TR}=m\widehat{RS}=m\widehat{ST}=x \). Then \( \frac{1}{3}(m\widehat{TR}+m\widehat{RS}+m\widehat{ST})=\frac{1}{3}(x + x+x)=x \). But \( m\widehat{TQ} \): since \( TQ \) is a segment from \( T \) to \( Q \), and \( TQ \) is the altitude of the equilateral triangle, the length of \( TQ=\frac{\sqrt{3}}{2}TR \), while the arc length formula \( s = r\theta \) (in radians). The angle for \( \widehat{TQ} \) (if we consider the arc from \( T \) to \( Q \) on some circle, but \( Q \) is the mid - point of \( RS \), not on a circle centered at \( T \) or \( R/S \) in the same way). So the formula \( m\widehat{TQ}=\frac{1}{3}(m\widehat{TR}+m\widehat{RS}+m\widehat{ST}) \) does not hold. So this statement cannot be concluded.

Answer:

\( m\widehat{TQ}=\frac{1}{3}(m\widehat{TR}+m\widehat{RS}+m\widehat{ST}) \) (the second option)