Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.1 AIME, harder Find the answer

3. A frog starts from point $A(-6,3)$ and jumps to point $B(-2,5)$ , then from point $B$ to point $C$ on the $y$ -axis, continues from point $C$ to point $D$ on the $x$ -axis, and finally returns from point $D$ to point $A$ . When the total distance of the frog's four jumps is minimized, the positions of points $C$ and $D$ are represented in a Cartesian coordinate system (保留作图痕迹), the method is:

(Note: "保留作图痕迹" translates to "保留作图痕迹" which means "retain the construction marks" in this context.)

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

3. As shown in Figure 5, the steps are as follows:
(1) Draw a Cartesian coordinate system, and points $A(-6,3)$ , $B(-2,5)$ ;
(2) Draw the point $A'$ , which is the reflection of point $A$ over the $x$ -axis;
(3) Draw the point $B'$ , which is the reflection of point $B$ over the $y$ -axis;
(4) Connect $A' B'$ to intersect the $y$ -axis at point $C$ , and the $x$ -axis at point $D$ . Then, $C$ and $D$ are the points required.

Note: By the principle that "the shortest distance between two points is a straight line," the minimum perimeter of quadrilateral $A B C D$ is
$A B + A' B' = 2 \sqrt{5} + 8 \sqrt{2}.$

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.