# Two parallel chords of lenght 24cm and 10cm which lie on opposite sides of the circle are 17cm apart.calculate the radius of the circle to the nearest whole number.

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## Did you make your sketch?

Let the distance from the longer chord to the circle be x
then the distance from the shorter chord to the circle is 17-x

then r^2 = x^2 + 12^2
and
r^2 = (17-x)^2 + 5^2

so x^2 + 144 = (17-x)^2 + 25

x^2 + 144 = 289 - 34x + x^2 + 25
34x = 170
x = 5

then r^2 = x^2 + 12^2
r^2 = 25+144 = 169

r = √169 = 13

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## To find the radius of the circle, we can use a property of circles that states that the two chords that are equidistant from the center of the circle are congruent.

Let's label the two parallel chords as AB and CD. We are given that the length of AB is 24 cm, the length of CD is 10 cm, and the distance between the two chords is 17 cm.

To find the radius, we need to find the distance from the center of the circle to one of the parallel chords.

Let's call the radius of the circle r.

From the center of the circle, draw a line perpendicular to AB and CD. This line will pass through the midpoints of AB and CD.

Let's label the midpoints of AB and CD as E and F, respectively.

Since AB and CD are parallel chords, the line EF is parallel to both chords.

We now have a triangle AEF, where AE = EF = r (since E is the midpoint of AB) and AF = 17/2 cm (as given).

Using the Pythagorean theorem, we can find the value of EF.

AF^2 = AE^2 + EF^2

(17/2)^2 = r^2 + (24/2)^2

(289/4) = r^2 + 144

289 = 4r^2 + 576

4r^2 = 289 - 576

4r^2 = -287

r^2 = -287/4

However, we can't have a negative value for r^2, so it seems that there was an error in the problem statement or calculation.

Please double-check the given information or provide additional details to proceed further.

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## To calculate the radius of the circle, we can make use of the properties of a circle and the given information.

Let's denote the radius of the circle as 'r'.

We are given two parallel chords of lengths 24cm and 10cm, which lie on opposite sides of the circle, and are 17cm apart.

Now, we can use a property of a circle that states: "If two chords of a circle are parallel, they cut off congruent arcs from the circle."

Since the chords are 17cm apart and the lengths of the chords are given as 24cm and 10cm, the longer chord will cut off an arc of length 24cm, and the shorter chord will cut off an arc of length 10cm.

To find the distance from the center of the circle to each chord, we can use the formula:

Distance from center to chord = 0.5 * sqrt(2 * r^2 - c^2)

where 'r' is the radius of the circle and 'c' is the length of the chord.

For the longer chord (24cm), the distance from the center of the circle to the chord would be:

Distance_1 = 0.5 * sqrt(2 * r^2 - 24^2)

For the shorter chord (10cm), the distance from the center of the circle to the chord would be:

Distance_2 = 0.5 * sqrt(2 * r^2 - 10^2)

Since the chords are parallel and lie on opposite sides of the circle, the sum of the distances from the center of the circle to each chord would be equal to the distance between the chords, which is 17cm:

Distance_1 + Distance_2 = 17

Substituting the calculated values for Distance_1 and Distance_2, we get:

0.5 * sqrt(2 * r^2 - 24^2) + 0.5 * sqrt(2 * r^2 - 10^2) = 17

Simplifying the equation and solving for 'r', we would get the value of the radius of the circle to the nearest whole number.