Question

if an inscribed angle intercepts an arc of 80°, what is the measure of the inscribed angle?
a. 40°
b. 60°
c. 90°
d. 20°

if two inscribed angles intercept the same arc, what can you say about the angles?
a. one is half the other.
b. one is twice the other.
c. they are unrelated.
d. they are congruent.

what do you call a line that touches a circle at exactly one point?
a. radius
b. chord
c. diameter
d. tangent

what is the measure of an inscribed angle that intercepts a 160° arc?
a. 80°
b. 40°
c. 100°
d. 120°

if a circle has a radius of 10 cm and a chord 12 cm long, what is the perpendicular distance from the center to the chord?
a. 10 cm
b. 6 cm
c. 9 cm
d. 8 cm

if two chords are equidistant from the center of a circle, what can you conclude about their lengths?
a. they are perpendicular to each other.
b. they are congruent.
c. one is twice as long as the other.
d. one is half as long as the other.

Explanation:

First question:

Step1: Recall the inscribed - angle theorem

The measure of an inscribed angle is half the measure of its intercepted arc. Let the measure of the inscribed angle be \(I\) and the measure of the intercepted arc be \(A\). The formula is \(I=\frac{1}{2}A\).

Step2: Substitute the value of the arc

Given \(A = 80^{\circ}\), then \(I=\frac{1}{2}\times80^{\circ}=40^{\circ}\).

Second question:

Step1: Use the inscribed - angle theorem

If two inscribed angles \(\angle1\) and \(\angle2\) intercept the same arc \(A\). By the formula \(I=\frac{1}{2}A\), for \(\angle1\), \(I_1=\frac{1}{2}A\) and for \(\angle2\), \(I_2=\frac{1}{2}A\). So \(\angle1=\angle2\).

Third question:

Step1: Recall the definitions

• A radius is a line segment from the center of a circle to a point on the circle.
• A chord is a line segment with both endpoints on the circle.
• A diameter is a chord that passes through the center of the circle.
• A tangent is a line that touches a circle at exactly one point.

Fourth question:

Step1: Apply the inscribed - angle formula

Let the measure of the inscribed angle be \(I\) and the measure of the intercepted arc be \(A = 160^{\circ}\). Using \(I=\frac{1}{2}A\), we get \(I=\frac{1}{2}\times160^{\circ}=80^{\circ}\).

Fifth question:

Step1: Use the property of chords and the Pythagorean theorem

Let the radius of the circle be \(r = 10\mathrm{cm}\), the length of the chord be \(l = 12\mathrm{cm}\). The perpendicular from the center of the circle to a chord bisects the chord. So the half - length of the chord \(x=\frac{l}{2}=\frac{12}{2}=6\mathrm{cm}\).

Step2: Apply the Pythagorean theorem

Let the perpendicular distance from the center to the chord be \(d\). Using the Pythagorean theorem \(d=\sqrt{r^{2}-x^{2}}\). Substitute \(r = 10\mathrm{cm}\) and \(x = 6\mathrm{cm}\), we have \(d=\sqrt{10^{2}-6^{2}}=\sqrt{100 - 36}=\sqrt{64}=8\mathrm{cm}\).

Sixth question:

Step1: Use the property of chords in a circle

If two chords are equidistant from the center of a circle, then by the property of circles, the lengths of the chords are equal (congruent).

Answer:

1. a. \(40^{\circ}\)
2. d. They are congruent.
3. d. Tangent
4. a. \(80^{\circ}\)
5. d. \(8\mathrm{cm}\)
6. b. They are congruent.