Example

Title: Circles

Grade: 1400-a Lesson: S3-L1

Explanation: Hello Students, time to learn examples. Let us take turns and read each example. Explain each step. Pay special attention to steps and pictures and communicate in your own words.

Examples:

The circumference of a circle is 31.4 cm. Find its radius.

Step 1a

Given,

The circumference of a circle = 31.4 cm

We know that the circumference(c) of a circle = 2πr.

Let the radius of the circle be r.

Explanation: We know that the circle’s circumference formula is \$2 \pi r\$. Here we have to find the radius(r) using the circumference value C = 31.4 cm.

Step 1b

Substitute the values in the formula

31.4 = 2πr

Divide both sides of the equation by 2π

\$(31.4) / (2π) = r\$

\$\cancel(31.4)^10 / (2 \times \cancel(3.14)^1) = r\$

\$\cancel(10)^5 / \cancel2^1 = r\$

r = 5 cm

Explanation: Here, we substitute the 'C' value in the formula. To find the circle’s radius, we substitute the π = 3.14. On cancellation, we get the circle’s radius is equal to 5cm.

An arc length in a circle is 6 cm, and the central angle corresponding to the arc is 40°. Find the radius of the circle.

Step 2a

The given arc length is 6 cm and the central angle is 40°

To find the radius of the circle, we can use the formula relating the arc length,

central angle, and radius:

arc length = radius \$times\$ central angle

Explanation: Here we have to find the radius(r) of the circle by using the arc length and central angle
arc length = radius \$times\$ central angle

Step 2b

Substituting these values into the formula, we have: 6 cm = r × 40°

The angle to be in radians rather than°. Since 1 radian is equal to

\$(180°)/ π\$, we can convert the angle to radians.

⇒ \$6 cm = r \times ((40°) times (π / 180°))\$

⇒ \$6 = r \times ((2π) / 9) \$ radians

⇒ \$6 / ((2π) / 9) = r\$ radians

⇒ \$r = (54 cm) / π\$ radians

⇒ \$r = (54 cm) / 3.14\$ radians

Thus, the radius of the circle is 17.1828 cm.

Explanation: Here, we substitute the given values in the formula. To find the circle’s radius, we substitute the π = 3.14. On solving, we get the radius of the circle is 17.1828cm.

The equation \$x^2 + y^2 -6x + 8y = 5\$ is that of a circle in the xy-plane. What is the circle’s diameter?

Step 3a

The given equation \$"x"^2 + "y"^2 - 6"x" + 8"y" = 5\$
Rearrange the terms to group the x terms and the y terms:
\$("x"^2 - 6"x") + ("y"^2 + 8"y") = 5\$

Explanation: In this step, the given equation \$"x"^2 + "y"^2 - 6"x" + 8"y" = 5\$ is rearranged to group the x terms and the y terms: \$("x"^2 - 6"x") + ("y"^2 + 8"y") = 5\$.

Step 3b

Complete the square for x by adding and subtracting \$(6/2)^2 = 9\$.
Similarly, complete the square for y by adding and subtracting
⇒ \$(8/2)^2 = 16\$
⇒ \$("x"^2 - 6"x" + 9) + ("y"^2 + 8"y" + 16) = 5 + 9 + 16 \$
⇒ \$("x" - 3)^2 + ("y" + 4)^2 = 30\$

Explanation: In this step, Complete the square for x and y by adding and subtracting \$(6/2)^2 = 9\$ and \$(8/2)^2 = 16\$, respectively. Simplify the equation to \$("x" - 3)^2 + ("y" + 4)^2 = 30\$.

Step 3c

Compare the equation with the standard form of a circle equation:
\$("x" - "h")^2 + ("y" - "k")^2 = "r"^2\$,
we find: h = 3, y = -4, and \$"r"^2 = 30\$
So, the radius r of the circle is \$\sqrt 30\$

Explanation: Equation compared to standard circle form: \$(x-h)^2 + (y-k)^2 = r^2\$.
Results: h = 3, y = -4, \$"r"^2\$ = 30. Radius r = \$sqrt30\$.

Step 3d

The diameter of the circle is twice the radius. Therefore, the diameter d is:
⇒ d = 2r
⇒ d = \$2\sqrt30\$
So, the diameter of the circle is \$2\sqrt30\$ units.

Explanation: The diameter d is calculated using the formula d = 2r. If the radius is \$sqrt30\$, the circle’s diameter is \$2\sqrt30\$ units.


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