The radius of a circle is 8cm, and the length of one of its chords is 12cm. Find the distance of the chord from the centre.
You can create a right-angled triangle
with hypotenuse = 8 , (the radius)
base of 6 , (1/2 your chord)
and the height unknown, call it x
x^2 + 6^2 = 8^2
x^2 = 28
x = √28 = 2√7
The radius of a circle is 13 cm and the lenth of one of its chords is 10 cm. Find the distance of the chord from the centre
To find the distance of the chord from the center of the circle, we can start by drawing the circle and its chord. Let's call the center of the circle point O, the chord AB, and the point where the chord intersects the circle point C.
We know that the radius of the circle (OA or OB) is 8 cm. Since the chord AB intersects the center of the circle (point O), it divides the chord into two equal parts.
Now, let's divide the chord AB into two equal parts, and label the midpoint M. So, AM = MB = 6 cm.
Next, draw the perpendicular bisector of chord AB. The perpendicular bisector of a chord passes through the center of the circle.
Since AM = MB = 6 cm, the perpendicular bisector will pass through point M and point O, the center of the circle.
Now, we have formed a right-angled triangle OMC, where OC is the perpendicular bisector and OM is the radius.
Let's use the Pythagorean theorem to find the length of OC.
The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In triangle OMC, OM represents the radius (8 cm) and MC is the distance of the chord from the center (which we want to find).
Using the Pythagorean theorem, we can write:
OC^2 = OM^2 + MC^2
We know that OM is 8 cm, so we can substitute this value:
OC^2 = (8 cm)^2 + MC^2
OC^2 = 64 cm^2 + MC^2
We also know that AM is half the length of the chord (12 cm) which is 6 cm. This helps us deduce that MC is half of AM, or 3 cm.
Substituting MC = 3 cm:
OC^2 = 64 cm^2 + (3 cm)^2
OC^2 = 64 cm^2 + 9 cm^2
OC^2 = 73 cm^2
To find OC, we take the square root of both sides of the equation:
OC = sqrt(73 cm^2)
OC ≈ 8.54 cm
Therefore, the distance of the chord from the center of the circle is approximately 8.54 cm.