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Sagot :
Certainly! Let's analyze the problem step-by-step.
We need to find the distance between two parallel chords, [tex]\(XY\)[/tex] and [tex]\(RS\)[/tex], in a circle with a radius of [tex]\(13\)[/tex] cm. The chords [tex]\(XY\)[/tex] and [tex]\(RS\)[/tex] are on opposite sides of the center of the circle and have lengths [tex]\(24\)[/tex] cm and [tex]\(20\)[/tex] cm, respectively.
To do this, we will use the fact that the perpendicular distance from the center of the circle to a chord can be found using the Pythagorean theorem.
### Step-by-Step Solution
1. Determine the distance from the center to the chord [tex]\(XY\)[/tex]:
- The radius of the circle [tex]\(r\)[/tex] is [tex]\(13\)[/tex] cm.
- The length of the chord [tex]\(XY\)[/tex] is [tex]\(24\)[/tex] cm.
Let's denote the distance from the center of the circle to the chord [tex]\(XY\)[/tex] as [tex]\(d_{XY}\)[/tex].
The perpendicular distance from the center to the chord forms a right triangle with half of the chord as one leg and the radius as the hypotenuse.
Using the Pythagorean theorem:
[tex]\[ r^2 = \left(\frac{XY}{2}\right)^2 + d_{XY}^2 \][/tex]
Plugging in the values:
[tex]\[ 13^2 = \left(\frac{24}{2}\right)^2 + d_{XY}^2 \][/tex]
[tex]\[ 169 = 12^2 + d_{XY}^2 \][/tex]
[tex]\[ 169 = 144 + d_{XY}^2 \][/tex]
[tex]\[ d_{XY}^2 = 169 - 144 \][/tex]
[tex]\[ d_{XY}^2 = 25 \][/tex]
[tex]\[ d_{XY} = \sqrt{25} \][/tex]
[tex]\[ d_{XY} = 5 \text{ cm} \][/tex]
2. Determine the distance from the center to the chord [tex]\(RS\)[/tex]:
- The radius of the circle [tex]\(r\)[/tex] is [tex]\(13\)[/tex] cm.
- The length of the chord [tex]\(RS\)[/tex] is [tex]\(20\)[/tex] cm.
Let's denote the distance from the center of the circle to the chord [tex]\(RS\)[/tex] as [tex]\(d_{RS}\)[/tex].
Using the same approach:
[tex]\[ r^2 = \left(\frac{RS}{2}\right)^2 + d_{RS}^2 \][/tex]
Plugging in the values:
[tex]\[ 13^2 = \left(\frac{20}{2}\right)^2 + d_{RS}^2 \][/tex]
[tex]\[ 169 = 10^2 + d_{RS}^2 \][/tex]
[tex]\[ 169 = 100 + d_{RS}^2 \][/tex]
[tex]\[ d_{RS}^2 = 169 - 100 \][/tex]
[tex]\[ d_{RS} = \sqrt{69} \][/tex]
[tex]\[ d_{RS} \approx 8.31 \text{ cm} \][/tex]
3. Calculate the distance between the two chords:
Both chords are on opposite sides of the center. So, the total distance between the two chords is the sum of the distances from the center to each chord.
[tex]\[ \text{Distance between chords} = d_{XY} + d_{RS} \][/tex]
[tex]\[ \text{Distance between chords} = 5 + 8.31 \][/tex]
[tex]\[ \text{Distance between chords} \approx 13.31 \text{ cm} \][/tex]
Therefore, the distance between the two chords [tex]\(XY\)[/tex] and [tex]\(RS\)[/tex] is approximately [tex]\(13.31\)[/tex] cm.
We need to find the distance between two parallel chords, [tex]\(XY\)[/tex] and [tex]\(RS\)[/tex], in a circle with a radius of [tex]\(13\)[/tex] cm. The chords [tex]\(XY\)[/tex] and [tex]\(RS\)[/tex] are on opposite sides of the center of the circle and have lengths [tex]\(24\)[/tex] cm and [tex]\(20\)[/tex] cm, respectively.
To do this, we will use the fact that the perpendicular distance from the center of the circle to a chord can be found using the Pythagorean theorem.
### Step-by-Step Solution
1. Determine the distance from the center to the chord [tex]\(XY\)[/tex]:
- The radius of the circle [tex]\(r\)[/tex] is [tex]\(13\)[/tex] cm.
- The length of the chord [tex]\(XY\)[/tex] is [tex]\(24\)[/tex] cm.
Let's denote the distance from the center of the circle to the chord [tex]\(XY\)[/tex] as [tex]\(d_{XY}\)[/tex].
The perpendicular distance from the center to the chord forms a right triangle with half of the chord as one leg and the radius as the hypotenuse.
Using the Pythagorean theorem:
[tex]\[ r^2 = \left(\frac{XY}{2}\right)^2 + d_{XY}^2 \][/tex]
Plugging in the values:
[tex]\[ 13^2 = \left(\frac{24}{2}\right)^2 + d_{XY}^2 \][/tex]
[tex]\[ 169 = 12^2 + d_{XY}^2 \][/tex]
[tex]\[ 169 = 144 + d_{XY}^2 \][/tex]
[tex]\[ d_{XY}^2 = 169 - 144 \][/tex]
[tex]\[ d_{XY}^2 = 25 \][/tex]
[tex]\[ d_{XY} = \sqrt{25} \][/tex]
[tex]\[ d_{XY} = 5 \text{ cm} \][/tex]
2. Determine the distance from the center to the chord [tex]\(RS\)[/tex]:
- The radius of the circle [tex]\(r\)[/tex] is [tex]\(13\)[/tex] cm.
- The length of the chord [tex]\(RS\)[/tex] is [tex]\(20\)[/tex] cm.
Let's denote the distance from the center of the circle to the chord [tex]\(RS\)[/tex] as [tex]\(d_{RS}\)[/tex].
Using the same approach:
[tex]\[ r^2 = \left(\frac{RS}{2}\right)^2 + d_{RS}^2 \][/tex]
Plugging in the values:
[tex]\[ 13^2 = \left(\frac{20}{2}\right)^2 + d_{RS}^2 \][/tex]
[tex]\[ 169 = 10^2 + d_{RS}^2 \][/tex]
[tex]\[ 169 = 100 + d_{RS}^2 \][/tex]
[tex]\[ d_{RS}^2 = 169 - 100 \][/tex]
[tex]\[ d_{RS} = \sqrt{69} \][/tex]
[tex]\[ d_{RS} \approx 8.31 \text{ cm} \][/tex]
3. Calculate the distance between the two chords:
Both chords are on opposite sides of the center. So, the total distance between the two chords is the sum of the distances from the center to each chord.
[tex]\[ \text{Distance between chords} = d_{XY} + d_{RS} \][/tex]
[tex]\[ \text{Distance between chords} = 5 + 8.31 \][/tex]
[tex]\[ \text{Distance between chords} \approx 13.31 \text{ cm} \][/tex]
Therefore, the distance between the two chords [tex]\(XY\)[/tex] and [tex]\(RS\)[/tex] is approximately [tex]\(13.31\)[/tex] cm.
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