Question

1. in the circle shown below, \\(\overline{qt}\\) and \\(\overline{ru}\\) are diameters. the measures of arcs \\(qr\\) and \\(st\\) are given. determine the measure of the following arcs: image of a circle with center c, points q, r, s, t, u, p; arc qr is 46°, arc st is 67° 11a arc ut measure = enter your next step here

Explanation:

Step1: Note diameter property

Diameter subtends 180° arc: $\widehat{QT} = 180^\circ$, so $\widehat{QR} + \widehat{RS} + \widehat{ST} = 180^\circ$. Given $\widehat{QR}=46^\circ$, $\widehat{ST}=67^\circ$, so $\widehat{RS}=180-46-67=67^\circ$.

Step2: Use full circle sum

Total circle arc: $360^\circ$. Arcs: $\widehat{QR}=46^\circ$, $\widehat{RS}=67^\circ$, $\widehat{ST}=67^\circ$, let $\widehat{PQ}=x$, $\widehat{PU}=y$, $\widehat{UT}=z$. $\widehat{RU}$ is diameter: $\widehat{RS} + \widehat{ST} + \widehat{TU} + \widehat{UP} + \widehat{PQ} + \widehat{QR} = 360^\circ$? No—$\widehat{RU}$ is diameter, so $\widehat{RSU}=180^\circ$? Wait, simpler: $\widehat{QT}=180^\circ$ (diameter), $\widehat{RU}=180^\circ$ (diameter). $\widehat{RU}$ includes $\widehat{RS} + \widehat{ST} + \widehat{TU}$? No, $\widehat{RU}$ is from R to U: $\widehat{RS} + \widehat{ST} + \widehat{TU} + \widehat{UP} + \widehat{PQ} + \widehat{QR}$ no—wait, $\widehat{QT}=180^\circ = \widehat{QR}+\widehat{RS}+\widehat{ST}=46+67+67=180$, correct. Then $\widehat{PUQ}$ is the other half: $360-180=180^\circ$. $\widehat{RU}$ is diameter: $\widehat{RTU}$? No, $\widehat{RU}=180^\circ = \widehat{RS}+\widehat{ST}+\widehat{TU}$? Wait $\widehat{RS}+\widehat{ST}=67+67=134$, so $\widehat{TU}=180-134=46^\circ$.

Step3: Confirm arc UT

$\widehat{UT}=46^\circ$ (since $\widehat{RU}=180^\circ = \widehat{RS}+\widehat{ST}+\widehat{UT}$).

Answer:

46