Mada za sehemu hiiCirclesMada 5
- Definition of terms
- Central angle
- Angles properties
- Chord properties of a circle
- Tangent
Imagine that you are on one side of a perfectly circular lake and looking across to a fishing pier on the other side. The chord is the line going across the circle from point A (you) to point B (the fishing pier). The circle outlining the lake's perimeter is called the circumference. A chord of a circle is a line that connects two points on a circle's circumference.
To illustrate further, let's look at several points of reference on the same circular lake from before. If each point of reference (i.e., duck feeding area, picnic tables, you, water fountain, and fishing pier) were directly on this lake's circumference, then each line connecting a point to another point on the circle would be chords.
The line between the fishing pier and you is now chord AC. The line between the water fountain and duck feeding area is now chord BE. The line between you and the picnic tables is chord CD.
If we had a chord that went directly through the center of a circle, it would be called a diameter. If we had a line that did not stop at the circle's circumference and instead extended into infinity, it would no longer be a chord; it would be called a secant.
In a circle, the perpendicular from the center of the circle to a chord bisects the chord. This means that the line segment from the center of the circle to the chord divides the chord into two equal parts, and the line is perpendicular to the chord at the point of bisection.
Proof of theorem
Let the center of the circle be O, and let the chord be AB. Drop a perpendicular from the center O to the chord AB, meeting it at point E (where E is the midpoint of AB).
Since OE is perpendicular to AB and O is the center of the circle, it follows by the properties of the circle that OA = OB (radii of the circle). Also, by the perpendicular bisector property, we have AE = EB, which proves that the perpendicular from the center bisects the chord.
Parallel chords in the same circle always cut congruent arcs. Parallel chords intercept congruent arcs.
Construct a diameter perpendicular to the parallel chords. What does this diameter do to each chord? The diameter bisects each chord. Reflect across the diameter (or fold on the diameter). What happens to the endpoints? The reflection takes the endpoints on one side to the endpoints on the other side. It, therefore, takes arc to arc. Distances from the center are preserved.
What have we proven? Arcs between parallel chords are congruent.
Example 1:
The figure is a circle with center O. Given PQ = 12 cm, find the length of PA.
Solution:
The radius OB is perpendicular to PQ. So, OB is a perpendicular bisector of PQ.
This means that PA = AQ because the perpendicular bisector divides the chord into two equal parts. Therefore, PA = 12 cm / 2 = 6 cm.
So, the length of PA is 6 cm.
Example 2:
The figure is a circle with center O and diameter 10 cm. Given PQ = 1 cm, find the length of RS.
Solution:
In this case, we are given the diameter of the circle, which is 10 cm, so the radius r = 10 cm / 2 = 5 cm.
The distance from the center O to the chord PQ is a perpendicular line that bisects PQ.
Let the perpendicular from O meet PQ at point M. Since PQ = 1 cm, we know PM = MQ = 0.5 cm.
Next, using the Pythagorean theorem in triangle OMP, we can find the length of RS.
Since OM is perpendicular to PQ and forms a right triangle, we can use the Pythagorean theorem:
OM² + PM² = OP²
OM² + (0.5)² = 5²
OM² + 0.25 = 25
OM² = 24.75
OM ≈ √24.75 ≈ 4.97 cm
Now, using the properties of the circle and the fact that the length of the chord RS will be twice the length of OM, we get:
RS = 2 × OM ≈ 2 × 4.97 ≈ 9.94 cm
So, the length of RS is approximately 9.94 cm.
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