two cord /AB/ 10cm and /CD/ 8cm are of a circle of radius 12cm find the distance of each of the chord from the centre
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let the distance from the longer chord to the center be h1
h1^2 + 5^2 = 12^2
h1^2 = 119
h1 = √119 = ....
find the other distance in the same way
To find the distance of each chord from the center of the circle, we can use the perpendicular bisector of each chord.
Step 1: Find the midpoint of chord AB.
To find the midpoint, we take the average of the endpoints.
Midpoint of chord AB = (A + B) / 2
Step 2: Find the midpoint of chord CD.
Midpoint of chord CD = (C + D) / 2
Step 3: Draw a line from the center of the circle to the midpoints of both chords.
The distance from the center to the midpoint of chord AB is the perpendicular distance of chord AB from the center of the circle.
Similarly, the distance from the center to the midpoint of chord CD is the perpendicular distance of chord CD from the center of the circle.
Step 4: Find the lengths of these perpendicular distances.
We can use the Pythagorean theorem to find the lengths of these perpendicular distances. Let's call the distance from the center to the midpoint of chord AB as h1, and the distance from the center to the midpoint of chord CD as h2.
h1 = sqrt(r^2 - (AB/2)^2)
h2 = sqrt(r^2 - (CD/2)^2)
Where r is the radius of the circle.
Given that r = 12 cm, AB = 10 cm, and CD = 8 cm, we can substitute these values into the formulas to find the distances.
h1 = sqrt((12^2) - (10/2)^2)
= sqrt(144 - 25)
= sqrt(119)
≈ 10.92 cm
h2 = sqrt((12^2) - (8/2)^2)
= sqrt(144 - 16)
= sqrt(128)
≈ 11.31 cm
Therefore, the distance of chord AB from the center is approximately 10.92 cm, and the distance of chord CD from the center is approximately 11.31 cm.
To find the distance of each chord from the center of the circle, follow these steps:
Step 1: Draw the circle and label its center as point O.
Step 2: Draw the two chords AB and CD within the circle.
Step 3: To find the distance of chord AB from the center, drop a perpendicular line from the center O to AB. Let the intersection point be E.
Step 4: Since the perpendicular bisects the chord AB into two equal parts, each part measures half of the chord length. Therefore, AE = BE = 10/2 = 5 cm.
Step 5: Observe that ΔOEA is a right triangle with hypotenuse OE as the radius of the circle and AE as the perpendicular side. Use the Pythagorean theorem to find OE.
Using Pythagorean theorem: (OE)^2 = (OA)^2 - (AE)^2
= (12 cm)^2 - (5 cm)^2
= 144 cm^2 - 25 cm^2
= 119 cm^2
Taking the square root of both sides, we get:
OE = √(119 cm^2) ≈ 10.92 cm
Therefore, the distance of chord AB from the center is approximately 10.92 cm.
Step 6: Repeat steps 3-5 for chord CD.
Since chord CD is shorter than the radius, the perpendicular line dropped from the center O will not intersect CD within the circle. However, it will intersect CD at a point outside the circle. Let the intersection point be F.
Step 7: We can observe that ΔOFD is a right triangle with hypotenuse OF as the radius of the circle and FD as the perpendicular side.
Using Pythagorean theorem: (OF)^2 = (OD)^2 - (FD)^2
= (12 cm)^2 - (8 cm)^2
= 144 cm^2 - 64 cm^2
= 80 cm^2
Taking the square root of both sides, we get:
OF = √(80 cm^2) ≈ 8.94 cm
Therefore, the distance of chord CD from the center is approximately 8.94 cm.