Answer:
Step-by-step explanation:
1. A given chord on a circle is perpendicular to a radius through its center, and it is at a distance less that the radius of the circle.
2. A circle of center O has a radius of 13 units. If a chord AB of 10 units is drawn at a distance, d, to the center of the circle, determine the value of d.
3. From question 2, the radius = 13 units, length of chord = 10 units and distance of chord to center of the circle is d.
A radius that meet the chord at center C, and divides it into two equal parts.
So that;
AC = CB = 5 units
Applying Pythagoras theorem to ΔOCB,
OC = d, CB = 5 units and OB = 13 units
[tex]13^{2}[/tex] = [tex]5^{2}[/tex] + [tex]d^{2}[/tex]
169 = 25 + [tex]d^{2}[/tex]
169 - 25 = [tex]d^{2}[/tex]
144 = [tex]d^{2}[/tex]
⇒ d = [tex]\sqrt{144}[/tex]
= 12 units
Therefore, the chord is at a distance of 12 units to the center of the circle.
A chord is a straight line joining 2 points on the circumference of a circle.
The line that connects any two points of the circumference of a circle is known as a chord.
Theorem: A radius or diameter that is perpendicular to a chord divides the chord into two equal parts and vice versa.
Converse: The perpendicular bisector of a chord passes through the center of a circle.
Problem: In the given figure the radius of the circle is 5 cms. what is the length of the chord?
Solution:
Given to us,
radius of the circle, r = 5 cms,
RA = 1 cm,
Radius, r = OA = OP = OQ = 5 cms.
Also, OR = OA - OR = 4 cms.
In ΔOPQ,
According to Pythagorus theorum,
OP² = OR² + RP²
5² = 4² + RP²
RP² = 25 -16 = 9
RP = 3,
We know,
A radius or diameter that is perpendicular to a chord divides the chord into two equal parts and vice versa.
A radius (OA) that is perpendicular to a chord divides the chord into two equal parts(RP = RQ).
Therefore, chord = RP + RQ = 3+3 = 6 cms.
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