Question

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. 6 cm b. 10 cm c. 9 cm d. 8 cm if a chord is 12 cm long and its perpendicular distance from the center is 5 cm, what is the radius of the circle? a. 7 cm b. 12 cm c. 7.81 cm d. 10.3 cm what is the longest chord in a circle? a. any chord b. the diameter c. a tangent line d. a chord perpendicular to a radius

Explanation:

Step1: Recall circle - chord property

A perpendicular from the center of a circle to a chord bisects the chord.

Step2: First question - find perpendicular distance

The radius $r = 10$ cm and the chord length $l=12$ cm. The perpendicular from the center to the chord bisects the chord, so the half - chord length $a=\frac{l}{2}=\frac{12}{2} = 6$ cm. Using the Pythagorean theorem in the right - triangle formed by the radius, half - chord, and the perpendicular distance $d$ from the center to the chord ($r^{2}=a^{2}+d^{2}$), we can solve for $d$. So $d=\sqrt{r^{2}-a^{2}}=\sqrt{10^{2}-6^{2}}=\sqrt{100 - 36}=\sqrt{64}=8$ cm.

Step3: Second question - find radius

The chord length $l = 12$ cm, so the half - chord length $a=\frac{l}{2}=6$ cm, and the perpendicular distance $d = 5$ cm. Using the Pythagorean theorem $r=\sqrt{a^{2}+d^{2}}=\sqrt{6^{2}+5^{2}}=\sqrt{36 + 25}=\sqrt{61}\approx7.81$ cm.

Step4: Third question - identify longest chord

By definition, the diameter of a circle is the longest chord as it passes through the center of the circle and its length is $2r$ (where $r$ is the radius), while other chords have lengths less than or equal to the diameter.

Answer:

d. 8 cm
c. 7.81 cm
b. The diameter