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Math  /  Geometry

QuestionZain draws a circle with radius rr and center (h,k)(h, k) in the coordinate plane. He places the point (x,y)(x, y) on the circle. How can Zain use his drawing to derive the general equation of a circle in standard form? Use the drop-down menus to explain your answer.
Click the arrows to choose an answer from each menu. Using any center point (h,k)(h, k) and any point on the circle (x,y)(x, y), zain can draw a right triangle that has a hypotenuse of length rr land legs of lengths Choose... * Then, Zain can derive the general equation of a circle in standard form by applying the Choose...

Studdy Solution

STEP 1

1. The circle is drawn on a coordinate plane.
2. The center of the circle is at point (h,k)(h, k).
3. The radius of the circle is rr.
4. The point (x,y)(x, y) lies on the circle.

STEP 2

1. Identify the relationship between the center, a point on the circle, and the radius.
2. Use the Pythagorean Theorem to derive the equation of the circle.
3. Write the general equation of a circle in standard form.

STEP 3

Identify the relationship between the center, a point on the circle, and the radius:
- The distance from the center (h,k)(h, k) to the point (x,y)(x, y) is the radius rr. - This forms a right triangle with legs of lengths ∣x−h∣|x - h| and ∣y−k∣|y - k|.

STEP 4

Use the Pythagorean Theorem to derive the equation of the circle:
- According to the Pythagorean Theorem, the square of the hypotenuse (radius) is equal to the sum of the squares of the legs. - Therefore, r2=(x−h)2+(y−k)2r^2 = (x - h)^2 + (y - k)^2.

STEP 5

Write the general equation of a circle in standard form:
- The derived equation r2=(x−h)2+(y−k)2r^2 = (x - h)^2 + (y - k)^2 is the standard form of the equation of a circle.
The general equation of a circle in standard form is:
(x−h)2+(y−k)2=r2 (x - h)^2 + (y - k)^2 = r^2

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