Question

11 multiple choice 1 point
what is the diameter of a circle in which a chord 24 cm long is 5 cm from the center?
13 cm
26 cm
24 cm
10 cm
12 multiple choice 1 point
given the circle with center o and with ( m angle r o q = 160 ^ { circ } ), find the measure of minor arc ( widehat { p r } ).
( 210 ^ { circ } )
( 20 ^ { circ } )
( 40 ^ { circ } )
( 10 ^ { circ } )

Explanation:

Question 11

Step1: Use the property of chord and radius

The perpendicular from the center of a circle to a chord bisects the chord. So half - length of the chord \(l=\frac{24}{2}=12\mathrm{cm}\), and the distance from the center to the chord \(d = 5\mathrm{cm}\). Let the radius of the circle be \(r\).

Step2: Apply the Pythagorean theorem

By the Pythagorean theorem \(r=\sqrt{l^{2}+d^{2}}\). Substitute \(l = 12\) and \(d=5\) into the formula: \(r=\sqrt{12^{2}+5^{2}}=\sqrt{144 + 25}=\sqrt{169}=13\mathrm{cm}\).

Step3: Calculate the diameter

Since the diameter \(D = 2r\), and \(r = 13\mathrm{cm}\), then \(D=2\times13=26\mathrm{cm}\).

Step1: Use the property of a straight angle

We know that \(\angle POQ=180^{\circ}\) (a straight angle). Given \(\angle ROQ = 160^{\circ}\).

Step2: Calculate \(\angle POR\)

Since \(\angle POR+\angle ROQ=\angle POQ\), then \(\angle POR=\angle POQ-\angle ROQ\). Substitute \(\angle POQ = 180^{\circ}\) and \(\angle ROQ=160^{\circ}\) into the formula: \(\angle POR=180^{\circ}-160^{\circ}=20^{\circ}\).

Step3: Relate the central angle and the minor arc

The measure of a minor arc is equal to the measure of its corresponding central angle. So \(m\overset{\frown}{PR}=m\angle POR\)

Answer:

\(26\mathrm{cm}\)

Question 12