https://file.aiquickdraw.com/mathbot/file/1728355620099silkgoan.jpg

Let's break down each of the problems:

### Problem I
**Given:**
- Points \(P\) and \(Q\) are fixed in the Euclidean plane.
- \(R\) is a point moving such that \(|RS| = 6 \, \text{cm}\), where \(S\) is the midpoint of \(\overline{PQ}\).
- \(T\) is the midpoint of \(\overline{QR}\), and \(U\) is the point of intersection of \(\overline{PT}\) and \(\overline{RS}\).

**To find:** The locus of \(U\).

#### Solution Approach:
1. Let the coordinates of \(P\) and \(Q\) be fixed as \(P(x_1, y_1)\) and \(Q(x_2, y_2)\).
2. The midpoint \(S\) of \(\overline{PQ}\) can be calculated as:
   \[
   S\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)
   \]
3. Since \(|RS| = 6 \, \text{cm}\), the point \(R(x_r, y_r)\) must lie on a circle of radius 6 cm centered at \(S\).
4. The midpoint \(T\) of \(\overline{QR}\) can be found using:
   \[
   T\left(\frac{x_2 + x_r}{2}, \frac{y_2 + y_r}{2}\right)
   \]
5. To find the locus of \(U\), the point of intersection of the lines \(\overline{PT}\) and \(\overline{RS}\), we solve the equations of these two lines.

The final locus will depend on the parametric equations derived from these steps, and we expect the locus of \(U\) to be a conic section, likely a circle or ellipse, based on the movement of \(R\) constrained by \(|RS| = 6\).

### Problem II
**Given:**
- A circle \(c(O, r)\) with center \(O\) and radius \(r\).
- A chord \(\overline{AB}\) moving such that the length \(|AB|\) is constant.

**To find:** The locus of the midpoint of \(\overline{AB}\).

#### Solution Approach:
1. Let the equation of the circle be \(x^2 + y^2 = r^2\), with center at \(O(0, 0)\).
2. The length of the chord \(\overline{AB}\) is constant, say \(d\). The midpoint \(M\) of the chord will move as the chord rotates around the center.
3. Using the geometry of circles, the locus of the midpoint of a fixed-length chord in a circle is known to trace a **smaller circle** concentric with the original circle.

   The radius of this smaller circle can be calculated using the relationship:
   \[
   r_{\text{mid}} = \sqrt{r^2 - \left(\frac{d}{2}\right)^2}
   \]
   where \(d\) is the length of the chord and \(r\) is the radius of the original circle.

### Summary:
- For Problem I, the locus of \(U\) is expected to be a conic section based on the geometry of the intersections.
- For Problem II, the locus of the midpoint of the chord is a smaller circle with a radius derived from the given chord length.

Would you like further details on any step, or clarification on the solution paths? 

Here are 5 questions to expand upon this:
1. How do you derive the equation of the line \(\overline{PT}\) for Problem I?
2. Can you explain how to parametrize the motion of point \(R\) around \(S\)?
3. In Problem I, how can the intersection point \(U\) be found algebraically?
4. What is the role of the constant distance \(|RS|\) in shaping the locus of \(U\)?
5. How does the fixed chord length \( |AB| \) in Problem II relate to the radius of the smaller circle?

**Tip:** When solving locus problems, consider symmetry and geometric constraints to simplify the parametric equations or line intersections.

1. Let P and Q be two fixed points in the Euclidean plane, and let R be a point moving such that |RS| = 6 cm, where S is the midpoint of PQ. If T is the midpoint of QR, and U is the point of intersection of PT and RS, then find the locus of U.

2. Given a circle c(O, r) and a chord AB moving such that |AB| is a constant. Find the locus of the midpoint of AB.

Explore solutions to locus problems in Euclidean and circle geometry, including finding the locus of a point based on midpoints and intersecting lines. Learn to apply the midpoint formula, circle properties, and intersection theorems in advanced geometry problems.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation