People's Education Press: Junior High School Mathematics, Grade 8, Part 1: The Beauty of Stability – Exploring Triangle and Quadrilateral Properties Through Real-Life Examples

Hello, student. I'm your instructor for today. In this lesson, we'll begin Chapter 1 of 'Fundamental Properties of Triangles and Exploration of Polygons.' We encounter triangles not just in math textbooks, but also in suspension bridges, high-voltage towers, and even foldable doors at home—where geometric properties are put to the test. Today, we'll dive deep into why 'three' forms the foundation of stability, while 'four' represents the source of change.

Stability of Triangles vs. Instability of Quadrilaterals

The stability of a triangle stems from its geometric uniqueness: once the three side lengths are fixed, its shape and size are completely determined. This 'structural rigidity' makes it the soul of engineering construction. In contrast, quadrilaterals are inherently unstable—a flexible property that is equally essential in industrial designs requiring expansion or contraction.

Core Logic: From 'Stability' to 'Change'

Triangle (Stability): Once the three sides are fixed, their internal angles are also fixed. The shape cannot be altered unless the side lengths are compromised.
Quadrilateral (Instability): Even after the four side lengths are fixed, their internal angles can still vary. This characteristic is commonly eliminated by using diagonal bracing—converting the quadrilateral into two triangles.
Construction Condition: The sum of any two sides of a triangle must be greater than the third side. This rule is critical for determining whether wooden pieces can successfully form a closed structure.

Real-Life Example

On construction sites,Tower cranehas arms made of triangular trusses; whereas at school gates, the flexibility of quadrilaterals is used to createextendable barrier gates.

Common Pitfall Reminder

The diagonal brace added before the frame deforms is the key insight that transforms a quadrilateral into two triangles. When choosing the brace, ensure it satisfiesthe sum of two sides being greater than the third sidethe measurement principle.

🎯 Professor’s Message

Remember: triangle stability is a geometric constraint, while quadrilateral instability is a design freedom. This balance between 'stability' and 'motion' is what defines the beauty of engineering mechanics.

QUESTION 1

Which of the following shapes have stability? (1) Quadrilateral with diagonal, (2) Two adjacent quadrilaterals, (3) Pentagonal house shape, (4) Ordinary quadrilateral, (5) Ordinary pentagon, (6) Pentagon divided into triangles by diagonals

(1) and (4)

(1) and (6)

(2) and (5)

All have stability

QUESTION 2

Which of the following shapes has stability? ( )

A. Square

B. Rectangle

C. Right Triangle

D. Parallelogram

QUESTION 3

How many additional wooden braces are needed at minimum to prevent deformation of a quadrilateral frame (made of four rods)?

1 rod

2 rods

3 rods

4 rods

QUESTION 4

Given four wooden rods of lengths $10, 7, 5, 3$, how many ways can you choose three to form a triangle?

1 way

2 ways

3 ways

4 ways

QUESTION 5

Regarding the position of the altitude $AD$ in a triangle, which statement is correct?

Always inside the triangle

Always outside the triangle

Depends on the apex angle—it may lie inside, outside, or coincide with a side

Depends only on the base length

QUESTION 6

A polygon has an interior angle sum of $1260^\circ$. How many sides does it have?

Heptagon

Octagon

Nonagon

Decagon

QUESTION 7

Which regular polygon cannot tile the plane alone?

Regular triangle

Square

Regular pentagon

Regular hexagon

QUESTION 8

In $\triangle ABC$, $AD$ is a median, $AE$ is an altitude, $AE=2$ cm, and $S_{\triangle ABD}=1.5$ cm². What is the length of $BC$?

1.5 cm

3 cm

4.5 cm

6 cm

QUESTION 9

As shown in the diagram, where should the resort be built to maintain equal distance from three roads forming a triangle?

Intersection of the three altitudes

Intersection of the three medians

Intersection of the three angle bisectors

Intersection of the three perpendicular bisectors