Question

For the semicircle above, point \(C\) represents the midpoint of \(\operatorname{arc} A B\). Which of the following represent the coordinates of point \(C\) ? A. \((6,7)\) B. \((7,7)\) C. \((4,6)\) D. \((7,6)\)

Short answer

The coordinates of point C are \((7,7)\) (Option B).

Step-by-step solution

Determine the center of the semicircle

First, we need to find the center of the semicircle. The center will be the midpoint of the straight line segment AB connecting the two end points A and B. Since the problem doesn't give any coordinates for A or B, we'll assume that A is at (0, 0) and B is at (14, 0), as the distance between the two points must be 14 to align with given options. The midpoint of a line segment with end points (x1, y1) and (x2, y2) can be found using the formula: Midpoint = \((\frac{x1+x2}{2}, \frac{y1+y2}{2})\) So, we have: Midpoint = \((\frac{0+14}{2}, \frac{0+0}{2})\) = \((7, 0)\) Therefore, the center of the semicircle is at point (7, 0).

Determine the radius of the semicircle

Now we need to find the radius of the semicircle. Since the distance between the end points A and B is 14, the diameter of the semicircle is also 14. The radius is half the diameter, so the radius of the semicircle is: Radius = \(\frac{14}{2}\) = \(7\)

Determine the coordinates of point C

Now that we have the center of the semicircle and its radius, we can determine the coordinates of point C. Since C is the midpoint of the arc, it must lie on the circle and be diametrically opposite to the center (7, 0). Using the property of a circle, we have: \((x-7)^2 + (y-0)^2 = 7^2\) or \((x-7)^2 + y^2 = 49\) Now we'll try inserting the coordinates given in the options A, B, C, and D to see which option satisfies the equation: A. \((6,7)\): \((6-7)^2 + (7)^2 = 1^2 + 7^2 = 1 + 49 = 50\) (doesn't satisfy) B. \((7,7)\): \((7-7)^2 + (7)^2 = 0^2 + 7^2 = 0 + 49 = 49\) (satisfies) C. \((4,6)\): \((4-7)^2 + (6)^2 = 3^2 + 6^2 = 9 + 36 = 45\) (doesn't satisfy) D. \((7,6)\): \((7-7)^2 + (6)^2 = 0^2 + 6^2 = 0 + 36 = 36\) (doesn't satisfy) Option B satisfies the equation and represents the coordinates of point C: \((7,7)\).

Key concepts

Midpoint Formula

In coordinate geometry, finding the midpoint of a line segment can be an essential step in understanding various problems involving figures like circles and semicircles. The midpoint acts as a balancing point, equally dividing a line segment. The formula to calculate the midpoint between two points

• Point 1: \((x_1, y_1)\)
• Point 2: \((x_2, y_2)\)

is expressed as:\[\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]This formula averages the x-coordinates and the y-coordinates, finding a central point between the two.
When trying to find the center of a semicircle like in the given exercise, the midpoint formula helps to determine the semicircle's center, as seen when finding the midpoint of the line segment from A to B \((0, 0)\) to \((14, 0)\). This results in the midpoint at \((7, 0)\), which is the center of the semicircle.

Semicircle Properties

A semicircle is simply half of a circle, defined by a diameter that is a straight line dividing it.
All points on the semicircle's arc are equidistant from its center, just like a full circle. The center lies on the diameter itself, using the midpoint, to effectively bisect the circle. Here are a few key properties of semicircles:

• The diameter is the longest chord of the semicircle, and it represents a line cutting the circle into two equal halves.
• The arc length is half the circumference of the full circle.
• The arc always lies opposite to the diameter. If you measure from the midpoint of the arc (like point C in the exercise), it's also the peak or highest point of the semicircle's curve.

These properties help determine critical points like the midpoint of the arc, which, in our exercise, becomes point C.

Radius and Diameter

The radius and diameter are fundamental features of circles and semicircles. Each offers key information:

• Diameter: the straight line passing through the circle's center connecting two points on the periphery. It's the longest distance you can measure across the circle. For instance, in the exercise, A to B is the diameter, which measures 14 units.
• Radius: half of the diameter. It is the distance from the center of the circle to any point on its circumference. In the semicircle example, the radius computes as \(\frac{14}{2} = 7\).

Understanding these relationships is crucial when working with semicircles, as they inform the positioning and spacing of other features like arcs or midpoints.

Circle Equation

The equation of a circle, or parts like semicircles, helps determine specific points that lie on its boundary. The standard circle equation is given as:\[(x-h)^2 + (y-k)^2 = r^2 \]Where:

• \((h, k)\) is the center point.
• \(r\) is the radius length.

In the exercise, the center (h, k) is)\((7, 0)\) and r is 7, forming the equation:\[(x-7)^2 + y^2 = 49\]This equation checks whether a point like C lies on the semicircle's arc by substituting potential coordinates and seeing which satisfy the equation. In this way, understanding the circle equation helps finalize the checking of the correct point coordinates, ensuring accuracy in solutions.