Question

the chord of a circle is 12 centimeters long and cuts the circle segment whose height is 3 cm find the radius of the circle height of the segment equals the difference between the radius of the circle and the distance of the chord from the circle center​

Answer 1

Answer:

3is the radius of circle

the length of chord to centre is 6cm

Answer 2

Final Answer:

The radius of the circle where there is a 12 centimeters long chord of a circle cutting the circle segment with height is 3cm is 7.5cm.

Given:

The chord of a circle is 12 centimeters long and cuts the circle segment whose height is 3 cm.

The height of the segment equals the difference between the radius of the circle and the distance of the chord from the center of the circle.​

To Find:

The radius of the circle where there is a 12 centimeters long chord of a circle cutting the circle segment with height is 3 cm.

Explanation:

Notre the following points.

• The radius of the circle is the constant distance of the circumference of the circle from its center.
• The chord of the circle is any line segment whose endpoints are on the circumference of the circle.
• The segment of the circle is any part of the circumference of the circle.

• The distance of the chord from the center of the circle is perpendicular to the chord, bisects the chord and when extended touches the circle to become Its radius.
• The distance of the chord from the center of the circle extended to the circle, the other radius and the line joining these two radii, forms an isosceles triangle .
• The area of that isosceles triangle is equal to the half of the product of the radius and the half of the chord length.
• The area of that isosceles triangle is also equal to the half of the product of its base and the height.

Step 1 of 3

Assume the radius of the circle is r centimeter.

Thus, the isosceles triangle which is formed by the distance of the chord from the center of the circle extended to the circle, the other radius and the line joining these two radii has its base as the line joining those two radii.

The base of the isosceles triangle

$=\sqrt{6^2+3^2}\\ =\sqrt{45} cm$

Step 2 of 3

Now, the area of the isosceles triangle (formed by the distance of the chord from the center of the circle extended to the circle, the other radius and the line joining these two radii) is

= The half of the product of the radius and the half of the chord length

$=[\frac{1}{2} \times r\times 6]cm^2$

Again, the area of that isosceles triangle is

= The half of the product of the height and its base

$=[\frac{1}{2} \times \sqrt{r^2-\frac{45}{4} } \times \sqrt{45}]cm^2$

Step 3 of 3

Equate both the expressions of the area of that isosceles triangle to form the following equation and solve it

$\frac{1}{2} \times r \times 6=\frac{1}{2} \times \sqrt{r^2-\frac{45}{4} } \times \sqrt{45}\\ r \times 6= \sqrt{r^2-\frac{45}{4} } \times \sqrt{45}\\36r^2=45(r^2-\frac{45}{4})\\36r^2=45r^2-\frac{45^2}{4}\\45r^2-36r^2=\frac{45^2}{4}\\9r^2=\frac{45^2}{4}\\r=7.5cm$

Therefore, the required radius of the circle where there is a 12 centimeters long chord of a circle cutting the circle segment with height is 3cm is 7.5cm.

Know more from the following links.

https://brainly.in/question/631276

https://brainly.in/question/3526

#SPJ3