Lesson Example Discussion Quiz: Class Homework |
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
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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. |
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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
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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 |
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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
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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 |
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Explanation:
Here we have to find the radius(r) of the circle by using the arc length and central angle |
Step 2b
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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. |
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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
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The given equation \$"x"^2 + "y"^2 - 6"x" + 8"y" = 5\$ |
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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
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Complete the square for x by adding and subtracting \$(6/2)^2 = 9\$. |
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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
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Compare the equation with the standard form of a circle equation: |
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Explanation:
Equation compared to standard circle form: \$(x-h)^2 + (y-k)^2 = r^2\$. |
Step 3d
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The diameter of the circle is twice the radius. Therefore, the diameter d is: |
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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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