【People's Education Press】Junior High School Math, Grade 9, Semester 1: Relationship Determination: From Geometric Position to Quantitative Conversion of d and r

The core of geometric research lies in transforming 'intuitive positional relationships' into 'precise quantitative relationships.' This lesson aims to establish algebraic relationships between the distance from the center to a line (d) and the radius (r), enabling quantitative determination of the positional relationships between lines and circles or between two circles—forming the logical foundation for studying tangent properties.

Principle of Geometric-Quantitative Transformation

When determining the relationship between line $l$ and circle $\odot O$, the sole criterion is the comparison of the distance $d$ from the center to the line with the radius $r$:

Intersecting: $d < r$ $\iff$ 2 common points (the line is called a secant)
Tangent: $d = r$ $\iff$ 1 common point (the line is called a tangent)
Separate: $d > r$ $\iff$ 0 common points

Five Cases of Relative Position Between Two Circles

When determining the relationship between two circles, the standard is the sum and difference of the distance between centers $d$ and the radii $r_1, r_2$:

Core Formulas

Externally separated: $d > r_1 + r_2$

Externally tangent: $d = r_1 + r_2$

Intersecting: $r_1 - r_2 < d < r_1 + r_2$ ($r_1 \ge r_2$)

Internally tangent: $d = r_1 - r_2$ ($r_1 > r_2$)

One inside the other: $d < r_1 - r_2$ ($r_1 > r_2$)

🎯 Core Principle

The geometric definition of positional relationships fundamentally reflects the number of solutions to a system of equations. Deeply understanding the critical tangent state ($d = r$ or $d = r_1 \pm r_2$) is the logical turning point where positional relationships transition from 'separate' to 'intersecting'.

QUESTION 1

Given a circle with diameter $13\text{ cm}$, if the distance from the center to the line is $d = 4.5\text{ cm}$, what is the positional relationship between the line and the circle, and how many common points are there?

Intersecting, 2 common points

Tangent, 1 common point

Separate, 0 common points

Tangent, 2 common points

QUESTION 2

In right triangle $\triangle ABC$, $\angle C=90^\circ$, $AC=3$, $BC=4$. What is the positional relationship between circle $\odot C$ centered at point $C$ with radius $r=2.4$ and side $AB$?

Separate

Tangent

Intersecting

Uncertain

QUESTION 3

What is the coordinate of the point corresponding to $P(x, y)$ after rotating $90^\circ$ clockwise about the origin?

$(-y, x)$

$(y, -x)$

$(-x, -y)$

$(x, -y)$

QUESTION 4

In $\triangle ABC$, $\angle ABC=50^\circ$, $\angle ACB=75^\circ$. Point $O$ is the incenter. What is the measure of $\angle BOC$?

$125^\circ$

$117.5^\circ$

$115^\circ$

$130^\circ$

QUESTION 5

In right triangle $\triangle ABC$, $\angle C=90^\circ$, with side lengths $a, b, c$ satisfying $AC=6$, $BC=8$, $AB=10$. What is the inradius $r$ of this triangle?