1.5 Plane Geometry: Triangles, Circles, Polygons & Perimeter/Area
Key Takeaways
- Memorize common Pythagorean triples and special right triangle ratios (30-60-90 and 45-45-90) to solve geometric lengths without full calculations.
- Apply similarity properties: linear scale factor k yields area scale factor k^2 and volume scale factor k^3.
- Utilize circle theorems regarding inscribed angles, central angles, tangent lines, and secant lengths.
- Calculate regular polygon interior angles using (n-2)*180/n and exterior angles using 360/n.
Last updated: July 2026
Plane Geometry: Triangles, Circles, Polygons & Perimeter/Area
Geometry on the ACET emphasizes fast shape identification, spatial reasoning, area/volume ratios, and circle properties. Calculators are unavailable, so memorizing key geometric ratios saves time.
1. Angle Relationships & Parallel Lines
When two parallel lines $L_1 \parallel L_2$ are cut by a transversal $T$:
- Corresponding Angles are equal ($1 = 5$).
- Alternate Interior Angles are equal ($3 = 6$).
- Alternate Exterior Angles are equal ($2 = 7$).
- Consecutive Interior Angles are supplementary ($3 + 5 = 180^\circ$).
2. Triangles: Properties, Special Triangles & Area Formulas
Fundamental Triangle Inequalities & Angle Theorems
- Angle Sum: $\theta_1 + \theta_2 + \theta_3 = 180^\circ$.
- Exterior Angle Theorem: The measure of an exterior angle equals the sum of its two remote interior angles.
- Triangle Inequality Theorem: For sides $a, b, c$: $|a - b| < c < a + b$.
Special Right Triangles & Pythagorean Triples
\mathbf{45^\circ-45^\circ-90^\circ \text{ Triangle:}} & \quad \text{Legs } x, x \implies \text{Hypotenuse } x\sqrt{2} \\ \mathbf{30^\circ-60^\circ-90^\circ \text{ Triangle:}} & \quad \text{Short leg } x \implies \text{Long leg } x\sqrt{3}, \text{ Hypotenuse } 2x \end{aligned}$$ **High-Frequency ACET Pythagorean Triples:** - $3 : 4 : 5 \quad (6:8:10, \; 9:12:15, \; 12:16:20)$ - $5 : 12 : 13 \quad (10:24:26)$ - $8 : 15 : 17$ - $7 : 24 : 25$ ### Comprehensive Area Formulas for Triangles 1. Standard Formula: $A = \frac{1}{2} b h$ 2. SAS Trigonometric Formula: $A = \frac{1}{2} a b \sin C$ 3. Equilateral Triangle Area: $A = \frac{s^2 \sqrt{3}}{4}$ 4. **Heron's Formula (Given 3 sides $a, b, c$):** $$A = \sqrt{s(s - a)(s - b)(s - c)}, \quad s = \frac{a + b + c}{2}$$ --- ## 3. Polygon Properties & Interior/Exterior Angles For a convex polygon with $n$ sides: - **Sum of Interior Angles:** $S_{\text{int}} = (n - 2) \times 180^\circ$ - **Interior Angle of Regular Polygon:** $I = \frac{(n - 2) \times 180^\circ}{n}$ - **Sum of Exterior Angles:** $S_{\text{ext}} = 360^\circ$ - **Exterior Angle of Regular Polygon:** $E = \frac{360^\circ}{n}$ - **Number of Diagonals:** $D = \frac{n(n - 3)}{2}$ --- ## 4. Circle Theorems & Sector Measures ### Primary Circle Formulas - Circumference: $C = 2\pi r = \pi d$ - Area: $A = \pi r^2$ - Arc Length ($L$ for central angle $\theta^\circ$): $L = \frac{\theta}{360^\circ} \times 2\pi r$ - Sector Area ($A_{\text{sector}}$): $A_{\text{sector}} = \frac{\theta}{360^\circ} \times \pi r^2$ ### Core Circle Theorems 1. **Inscribed Angle Theorem:** An inscribed angle is half the central angle intercepting the same arc: $\theta_{\text{inscribed}} = \frac{1}{2} \theta_{\text{central}}$. 2. **Thales' Theorem:** An angle inscribed in a semicircle is always a right angle ($90^\circ$). 3. **Tangent Line Theorem:** A line tangent to a circle is perpendicular to the radius at the point of tangency. 4. **Intersecting Chords Theorem:** $PA \cdot PB = PC \cdot PD$. 5. **Secant-Tangent Theorem:** $PT^2 = PA \cdot PB$. --- ## Worked Step-by-Step ACET Exam Problems ### Problem 1: Sector Area and Shaded Segment Calculation *A circle has a radius of $6\text{ cm}$. A central angle of $60^\circ$ intercepts an arc. What is the area of the shaded segment formed between the chord and the arc?* **Step 1: Calculate total sector area.** $$A_{\text{sector}} = \frac{60^\circ}{360^\circ} \times \pi (6)^2 = \frac{1}{6} \times 36 \pi = 6\pi \text{ cm}^2$$ **Step 2: Calculate area of the subtended triangle.** Since the central angle is $60^\circ$ and two sides are radii ($r=6$), the triangle is equilateral with side length $s = 6$. $$A_{\text{triangle}} = \frac{s^2 \sqrt{3}}{4} = \frac{6^2 \sqrt{3}}{4} = \frac{36 \sqrt{3}}{4} = 9\sqrt{3} \text{ cm}^2$$ **Step 3: Subtract triangle area from sector area.** $$A_{\text{segment}} = A_{\text{sector}} - A_{\text{triangle}} = (6\pi - 9\sqrt{3}) \text{ cm}^2$$ --- ### Problem 2: Similar Triangles and Area Scale Factors *In $\triangle ABC$, a line parallel to base $BC$ intersects side $AB$ at $D$ and side $AC$ at $E$. If $AD = 3\text{ cm}$ and $DB = 5\text{ cm}$, find the ratio of the area of $\triangle ADE$ to the area of trapezoid $DBCE$.* **Step 1: Determine side ratio between similar triangles.** $\triangle ADE \sim \triangle ABC$. Side length $AB = AD + DB = 3 + 5 = 8\text{ cm}$. Scale factor $k = \frac{AD}{AB} = \frac{3}{8}$. **Step 2: Determine area ratio.** $$\frac{\text{Area}(\triangle ADE)}{\text{Area}(\triangle ABC)} = k^2 = \left(\frac{3}{8}\right)^2 = \frac{9}{64}$$ **Step 3: Calculate area ratio relative to trapezoid.** Let $\text{Area}(\triangle ABC) = 64x$, so $\text{Area}(\triangle ADE) = 9x$. $$\text{Area}(\text{Trapezoid } DBCE) = 64x - 9x = 55x$$ $$\frac{\text{Area}(\triangle ADE)}{\text{Area}(\text{Trapezoid } DBCE)} = \frac{9x}{55x} = \frac{9}{55}$$ --- ## ACET Speed Tactics & Traps Summary | Problem Type | Common Student Trap | ACET Speed Tactic | | :--- | :--- | :--- | | **Triangles** | Using Pythagorean theorem on non-right triangles | Check angle measures or use Law of Cosines | | **Similar Figures** | Assuming area scales linearly with side length $k$ | Area scales as $k^2$; volume scales as $k^3$ | | **Inscribed Angles** | Setting inscribed angle equal to arc measure | Inscribed angle is **half** the intercepted arc | | **Regular Polygons** | Confusing interior angle sum with single angle | Divide sum $(n-2)180^\circ$ by $n$ for regular shapes |Test Your Knowledge
What is the area of a regular hexagon with a side length of 4 cm?
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Test Your Knowledge
In a circle with radius 10 cm, two chords intersect at point P inside the circle. If the segments of one chord are 4 cm and 9 cm, and one segment of the second chord is 3 cm, what is the length of the second chord?
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How many diagonals can be drawn inside a regular octagon (8-sided polygon)?
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