Question

1. 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. 8 cm b. 6 cm c. 10 cm d. 9 cm 26. which of the following is always true about the opposite sides of a parallelogram? a. they form a right angle b. they are congruent to the adjacent sides c. they are perpendicular d. they are parallel and congruent 27. what characterizes the incircle of a triangle? a. it is the same as the circumcircle b. it passes through all three vertices c. it is inscribed within a triangle and touches all three sides d. it is equidistant from all vertices

Explanation:

Step1: Recall circle - chord property

The perpendicular from the center of a circle to a chord bisects the chord. So the length of half - chord is $\frac{12}{2}=6$ cm. The radius of the circle is $r = 10$ cm.

Step2: Apply Pythagorean theorem

Let the perpendicular distance from the center to the chord be $d$. Using the Pythagorean theorem in the right - triangle formed by the radius, half - chord and the perpendicular from the center to the chord ($r^{2}=d^{2}+(\frac{l}{2})^{2}$, where $r$ is the radius and $l$ is the length of the chord), we have $d=\sqrt{r^{2}-(\frac{l}{2})^{2}}$. Substituting $r = 10$ cm and $\frac{l}{2}=6$ cm, we get $d=\sqrt{10^{2}-6^{2}}=\sqrt{100 - 36}=\sqrt{64}=8$ cm.

Step3: Recall parallelogram properties

The opposite sides of a parallelogram are parallel and congruent. They do not form right - angles (except in special cases like rectangles), are not congruent to adjacent sides in general, and are not perpendicular.

Step4: Recall incircle of a triangle properties

The incircle of a triangle is inscribed within the triangle and touches all three sides. The circum - circle passes through all three vertices, and the incircle is not the same as the circum - circle. The point equidistant from all vertices is the circum - center, not related to the incircle.

Answer:

1. A. 8 cm
2. d. They are parallel and congruent
3. c. It is inscribed within a triangle and touches all three sides