Question

which statements are correct? check all that apply. a chord is sometimes a radius. a diameter is always a chord. a tangent is never a secant. a chord can be longer than a diameter of the same circle. a chord can be shorter than a radius of the same circle. a radius is always congruent to another radius of the same circle.

Explanation:

Step1: Analyze each statement

• Statement 1: A chord is sometimes a radius

A chord is a line segment with endpoints on the circle. A radius has one - endpoint at the center and the other on the circle. So, a chord can never be a radius.

• Statement 2: A diameter is always a chord

A diameter is a line segment that passes through the center of the circle and has endpoints on the circle. By the definition of a chord (a line segment with endpoints on the circle), a diameter is a chord.

• Statement 3: A tangent is never a secant

A tangent is a line that touches the circle at exactly one point. A secant is a line that intersects the circle at two points. So, a tangent can never be a secant.

• Statement 4: A chord can be longer than a diameter of the same circle

The diameter is the longest chord in a circle. So, a chord cannot be longer than a diameter.

• Statement 5: A chord can be shorter than a radius of the same circle

Let the radius of the circle be \(r\). The length of a chord \(l = 2\sqrt{r^{2}-d^{2}}\), where \(d\) is the distance from the center of the circle to the chord. When \(d=\sqrt{r^{2}-\frac{r^{2}}{4}}=\frac{\sqrt{3}r}{2}\), \(l = r\). When \(d\) is larger (but \(d\lt r\)), \(l\lt r\).

• Statement 6: A radius is always congruent to another radius of the same circle

By the definition of a circle, all radii of a given circle have the same length.

Answer:

B. A diameter is always a chord.
C. A tangent is never a secant.
E. A chord can be shorter than a radius of the same circle.
F. A radius is always congruent to another radius of the same circle.