find the measure of arc $\\widehat{db}$ in $\\odot p$.\n$m\\widehat{db} = \\square^{\\circ}$\n(simplify your…

Explanation:

Step1: Identify the given central angles

The image shows circle $P$ with central angles $\angle BPT = 90^{\circ}$ (indicated by the right angle symbol) and $\angle TPC = 127^{\circ}$.

Step2: Determine the measure of angle BPC

Since $B, P, D$ and $T, P, C$ are not necessarily straight lines, we use the property that the sum of central angles in a circle is $360^{\circ}$. However, the diagram shows $TC$ and $BD$ as diameters. If $TC$ is a diameter, then $\angle TPC + \angle CPD = 180^{\circ}$. If $BD$ is a diameter, then $\angle BPT + \angle TPD = 180^{\circ}$. Looking at the straight lines, $TC$ and $BD$ are diameters intersecting at $P$.

Step3: Calculate the measure of angle CPD

Since $TC$ is a straight line (diameter), the angles on one side sum to $180^{\circ}$. $$\angle CPD = 180^{\circ} - \angle TPC = 180^{\circ} - 127^{\circ} = 53^{\circ}$$

Step4: Calculate the measure of angle BPD

Since $BD$ is a straight line (diameter), the angles on one side sum to $180^{\circ}$. $$\angle BPD = 180^{\circ} - \angle BPT = 180^{\circ} - 90^{\circ} = 90^{\circ}$$

Step5: Relate central angle to arc measure

The measure of an arc is equal to the measure of its corresponding central angle. $$m\widehat{DB} = \angle DPB$$

Step6: Final calculation

From Step 4, we found that $\angle BPD = 90^{\circ}$. $$m\widehat{DB} = 90^{\circ}$$

Answer:

$m\widehat{DB} = 90^{\circ}$