Justify your answer why your chosen object illustrates hinge theorem. List them below with justification. Name of Object Justification 1 2 3 4 5 6 Activity 3: Perfect Match!
Activity 3: Perfect Match!
The Correct Answer and Explanation is:
Hinge Theorem Application – Objects and Justifications
The Hinge Theorem (also called the SAS Inequality Theorem) states:
If two triangles have two pairs of congruent sides, but the included angle of one is larger than the included angle of the other, then the third side of the triangle with the larger angle is longer than the third side of the triangle with the smaller angle.
Below are real-world objects that illustrate the hinge theorem:
| Name of Object | Justification |
|---|---|
| 1. Scissors | The blades are two sides of a triangle, and the screw is the vertex. Opening the scissors wider (larger included angle) makes the distance between the blade tips (third side) longer. |
| 2. Door Hinges | A door swinging wider (larger included angle between the door and wall) makes the distance from the edge of the door to the edge of the frame (third side) increase. |
| 3. Crane Arm | Cranes have two arms joined at a pivot. When the angle between the arms increases, the reach (third side) also increases. |
| 4. Compass Drawing Tool | The two legs represent two equal sides. Increasing the angle between them makes the arc radius (distance between pencil and point) longer. |
| 5. Mechanical Pliers | When pliers are opened wider (larger angle between handles), the gap between the tips increases, demonstrating the longer third side. |
| 6. Adjustable Tripod | When two legs of the tripod are the same length, increasing the angle between them spreads the tripod wider, lengthening the base (third side). |
Explanation
The Hinge Theorem is a geometric principle that connects real-life movements and mechanisms with triangle properties. It essentially says that if two triangles have two pairs of sides that are the same, the triangle with the wider angle between those sides will have a longer third side. This concept appears in various tools and objects we use daily.
Take scissors, for instance. Each blade and the handle form two equal-length sides. As we open the scissors, the angle at the screw hinge increases. As the angle increases, the tip-to-tip distance (the third side of the triangle) also increases, clearly showing the hinge theorem in action.
The same happens with door hinges. A door opening wider creates a larger angle between the wall and the door. Since the hinge and door edge are constant lengths, the increased angle results in a larger arc or reach—again, the third side is longer.
Crane arms, tripods, compasses, and pliers all exhibit this same relationship. Each has two components of equal or fixed length with a pivot between them. When the angle between them changes, the length of the span or opening changes accordingly.
Understanding how the hinge theorem works helps engineers and designers predict how mechanisms will move or open. This makes the theorem not just a mathematical concept but a powerful tool in practical applications. These objects help us visualize and apply the hinge theorem beyond the classroom and into real-world mechanics.
