Perimeter: Caliper Measurement

A Vernier caliper has two scales: a fixed main scale and a sliding Vernier scale. Together, they measure dimensions with a precision of $0.1\text{ mm}$.

How to read it:
1. Read the value on the main scale just to the left of the Vernier `0` mark.
2. Find the Vernier mark (0 to 10) that aligns perfectly with a main scale line.
3. Multiply that mark by $0.1\text{ mm}$ and add it to the main scale reading.

How it works (The Vernier Principle):
• Each division on the main scale is exactly $1\text{ mm}$.
• The Vernier scale has 10 divisions spanning exactly $9\text{ mm}$ on the main scale, meaning each Vernier division is $0.9\text{ mm}$.
• The difference between one main scale division ($1\text{ mm}$) and one Vernier division ($0.9\text{ mm}$) is exactly $0.1\text{ mm}$.
• When the caliper opens by a fraction $0.y\text{ mm}$, the Vernier mark numbered $y$ aligns perfectly with a main-scale line.

• Example (Measuring $12.3\text{ mm}$): The whole part is $12\text{ mm}$. The fractional part is $0.3\text{ mm}$. The 3rd Vernier mark is located $2.7\text{ mm}$ ($3 \times 0.9\text{ mm}$) to the right of the Vernier `0`. Since the caliper is open at $12.3\text{ mm}$, this mark reaches $12.3 + 2.7 = 15.0\text{ mm}$, aligning perfectly with the $15\text{ mm}$ line on the main scale.

Two triangles are congruent ($\cong$) if they have the exact same shape and size.
• SSS Theorem: If all three sides of one triangle are equal to the three sides of another, they are congruent.
• Ordering: The vertex order is crucial! $\triangle ABC \cong \triangle DEF$ means vertex $A$ maps to $D$, $B$ to $E$, and $C$ to $F$.

$\Delta ABC \cong \Delta DEF$ (SSS: $a=d, b=e, c=f$)

Using geometric theorems, we can prove properties of rectangles:

• Diagonals are Equal: In a rectangle $ABCD$ (all angles $90^\circ$, opposite sides equal):
1. Triangles $\triangle ABC$ and $\triangle BAD$ are right-angled triangles that share leg $AB$, and have $BC=AD$.
2. By the Pythagorean theorem, since their legs are equal, their hypotenuses (the diagonals) must be equal: $AC = \sqrt{AB^2 + BC^2} = \sqrt{AB^2 + AD^2} = BD$.
3. Therefore, the diagonals are equal: $AC = BD$.

• Converse Law (diagonals equal $\implies$ rectangle): If opposite sides are equal ($AB=CD, BC=DA$) and diagonals are equal ($AC=BD$):
1. Triangles $\triangle ABC$ and $\triangle CDA$ share $AC$, and have $AB=CD, BC=DA$. By SSS congruency, $\triangle ABC \cong \triangle CDA \implies \angle B = \angle D$.
2. Similarly, using diagonal $BD$, $\triangle ABD \cong \triangle CDB \implies \angle A = \angle C$.
3. Compare $\triangle ABC$ and $\triangle BAD$. They share $AB$, have $BC=AD$, and $AC=BD$. By SSS congruency, $\triangle ABC \cong \triangle BAD \implies \angle A = \angle B$.
4. All four angles are equal: $\angle A = \angle B = \angle C = \angle D$. Since the angles sum to $360^\circ$, each must be $360^\circ / 4 = 90^\circ$.

• Logical AND ($\land$): Represents when multiple conditions must hold at the same time (e.g. $AB=CD \land BC=DA$). To type it, enter \land and press Enter, or use the shortcut land, or select $\land$ from the keyboard panel.

• Not Equal ($\neq$): Represents unequal segments or values (e.g. $AB \neq BC$). To type it, enter \neq and press Enter, or use the shortcut neq, or select $\neq$ from the keyboard panel (press Shift to find it).