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Chapter 39: Problem 688

What is the probable locus of the midpoints of the radii of a given circle?

Short Answer

Expert verified
The locus of the midpoints of the radii of a given circle with center O(h, k) and radius r is a circle with the same center (O) and with a radius equal to half the given radius (r/2). The locus can be represented by the equation: \[(x-h)^2 + (y-k)^2 = \left(\frac{r}{2}\right)^2.\]

Step by step solution

01

Identify the given circle

We are given a circle with a fixed center, say O, and a fixed radius, say r. We can represent this circle with the equation \[(x-h)^2 + (y-k)^2 = r^2,\] where (h, k) are the coordinates of the center O.
02

Find the midpoint of a radius

Let A(x鈧�, y鈧�) be a point on the given circle, and OA be the radius. The midpoint M of the radius OA can be found using the midpoint formula: \[M\left(\frac{x_{1}+h}{2}, \frac{y_{1}+k}{2}\right).\]
03

Show that M lies on the circle with half the radius

We need to prove that the distance from M to the center O is half the distance from A to O (which is the given radius r). We can use the distance formula to find the distance between M and O: \[d(M, O) = \sqrt{\left(\frac{x_{1}+h}{2} - h\right)^{2} + \left(\frac{y_{1}+k}{2} - k\right)^{2}}.\] Now simplify the expression for the distance: \[d(M, O) = \sqrt{\left(\frac{x_{1}-h}{2}\right)^{2}+\left(\frac{y_{1}-k}{2}\right)^{2}} = \frac{1}{2}\sqrt{(x_{1}-h)^{2}+(y_{1}-k)^{2}}.\] Since A(x鈧�, y鈧�) lies on the circle, we have: \[(x_1-h)^2 + (y_1-k)^2 = r^2\] Taking the square root of both sides, we get: \[\sqrt{(x_1-h)^2 + (y_1-k)^2} = r\] Now, comparing the expressions of \(d(M, O)\) and the radius r, we find that: \[d(M, O) = \frac{1}{2}r\] Hence, M lies on a circle with the same center as the given circle (O) and a radius equal to half the given radius (r/2).
04

Determine the locus

The locus of the midpoints of the radii of the circle with center O and radius r is a circle with the same center (O) and with a radius equal to half the given radius (r/2). Therefore, the locus can be represented by the equation: \[(x-h)^2 + (y-k)^2 = \left(\frac{r}{2}\right)^2.\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Circle Equations
Understanding circle equations is essential for analyzing geometric shapes and their properties. In general, the equation of a circle in the Cartesian coordinate system is given by \[ (x-h)^2 + (y-k)^2 = r^2, \] where \((h, k)\) represents the center of the circle, and \(r\) is the radius. This form is known as the standard equation of a circle.

The equation is derived from the distance formula, which calculates the distance between two points. It states that all points \((x, y)\) lying on the circle are at a fixed distance \(r\) from the center \((h, k)\). When we plot the set of all points that satisfy this equation, we will have a perfect circle on the graph.

When dealing with problems, for example, finding the locus of the midpoints of the radii of a circle, knowing the circle equation is crucial. It helps to determine the relationship between the original circle and the resulting locus. The circle's radius and the position of its center directly influence the final equation of the locus.
Midpoint Formula
The midpoint formula is a convenient way to find the exact center point between two points on a Cartesian plane. If you have two points, \(A(x_1, y_1)\) and \(B(x_2, y_2)\), the midpoint, \(M\), of the line segment \(AB\) is calculated using the formula: \[ M\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right). \]

This formula simply averages the x-coordinates and the y-coordinates of the points \(A\) and \(B\) to find the coordinates of the midpoint \(M\). It is a direct application of the concept of averages to geometry. For example, in the context of finding the midpoint of a radius in a circle, this formula is vital. It allows students to easily calculate the position of the midpoint along any radii of the circle, which is a step required to solve problems involving the properties and locus of midpoints in circles.
Distance Formula
The distance formula is a way to determine the length of the segment connecting two points \((x_1, y_1)\) and \((x_2, y_2)\) on the Cartesian plane. It is derived from the Pythagorean theorem and is written as: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}. \]

This formula is fundamental when dealing with circle-related problems, such as proving that the midpoints of the radii of a given circle lie on another circle with half the original radius. By applying the distance formula, one can show that these midpoints are equidistant from the center and that this distance is half of the original circle's radius. The distance formula, midpoint formula, and the equation of a circle work together, allowing students to break down and solve complex geometric problems methodically and confidently.

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Most popular questions from this chapter

Two concentric circles have radii whose lengths are 2 in. and 6 in. Line \(\mathrm{m}\) is drawn, in the accompanying figure, tangent to the smaller circle, (a) Describe fully the locus of points equidistant from the two circles, (b) Describe fully the locus of points at a given distance \(d\) from line \(\mathrm{m}\). (c) How many points are there which satisfy the conditions given in both parts (a) and (b) if: (1) \(\mathrm{d}<2\) in.? (2) \(\mathrm{d}=2\) in.? (3) \(\mathrm{d}=6\) in.? (4) \(\mathrm{d}>6\) in.?

Describe the locus of points determined by the equation \(\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}=2 \mathrm{x}\). (Hint: Complete the square in x.)

(a) Write an equation of the locus of points whose distance from the origin is \(5 .\) (b) Determine whether the point \((-3,4)\) is on the locus.

(a) Describe the locus of points 2 units from the y-axis and write an equation of this locus, (b) Describe the locus of points equidistant from the points \(\mathrm{P}_{1}(-4,2)\) and \(\mathrm{P}_{2}(-4,6)\) and write an equation for this locus. (c) Find the number of points which satisfy both conditions stated in (a) and (b) and give the coordinates of each point.

Write an equation for the locus of points equidistant from \((3,3)\) and \((4,4)\).
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