Solutions and Area Formulas

🔺 Properties of Triangles: Solutions & Area Formulas

The Triangle Detective Story 🕵️

Imagine you’re a detective. Someone gives you clues about a triangle—maybe three sides, or two sides and an angle. Your job? Figure out EVERYTHING about that triangle. That’s what we’re learning today!

🌟 The Big Picture: Solving Oblique Triangles

What’s an oblique triangle? Any triangle that’s NOT a right triangle. No 90° angle anywhere!

Think of it like this: A right triangle is like a pizza slice with one perfectly square corner. An oblique triangle is like a slice from a weird-shaped pizza—no perfect corners.

Why do we need special methods? SOH-CAH-TOA only works for right triangles. For oblique triangles, we need two powerful tools:

• Law of Sines: connects sides to their opposite angles
• Law of Cosines: like an upgraded Pythagorean theorem

graph TD
    A["Triangle Problem"] --> B{What info do you have?}
    B --> C["SSS: Three Sides"]
    B --> D["SAS: Two Sides + Included Angle"]
    B --> E["ASA/AAS: Two Angles + One Side"]
    C --> F["Use Law of Cosines"]
    D --> F
    E --> G["Use Law of Sines"]

🎯 SSS Case Solution (Three Sides Given)

The Situation: You know all three sides (a, b, c) but NO angles.

The Method: Use the Law of Cosines rearranged to find angles!

The Formula

$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$

Think of it like this: If you have a triangle made of sticks, you can figure out how “open” each corner is just by measuring the sticks!

Example

Triangle with sides: a = 7, b = 8, c = 9

Step 1: Find angle A $\cos A = \frac{8^2 + 9^2 - 7^2}{2 \times 8 \times 9}$ $\cos A = \frac{64 + 81 - 49}{144} = \frac{96}{144} = 0.667$ $A = \cos^{-1}(0.667) ≈ 48.2°$

Step 2: Find angle B using same method $\cos B = \frac{a^2 + c^2 - b^2}{2ac} = \frac{49 + 81 - 64}{126} ≈ 0.524$ $B ≈ 58.4°$

Step 3: Find angle C $C = 180° - 48.2° - 58.4° = 73.4°$

✅ Done! We found all three angles from just the sides!

🎯 SAS Case Solution (Two Sides + Included Angle)

The Situation: You know two sides AND the angle BETWEEN them.

Why “included”? The angle must be the one that’s tucked between your two known sides—like the hinge of a door between the door and frame.

The Method

Step 1: Use Law of Cosines to find the third side $c^2 = a^2 + b^2 - 2ab \cos C$

Step 2: Use Law of Sines to find remaining angles

Example

Given: a = 5, b = 7, angle C = 42°

Find side c: $c^2 = 5^2 + 7^2 - 2(5)(7)\cos(42°)$ $c^2 = 25 + 49 - 70(0.743)$ $c^2 = 74 - 52.01 = 21.99$ $c ≈ 4.69$

Find angle A (using Law of Sines): $\frac{\sin A}{a} = \frac{\sin C}{c}$ $\sin A = \frac{5 \times \sin(42°)}{4.69} = \frac{5 \times 0.669}{4.69} ≈ 0.713$ $A ≈ 45.5°$

Find angle B: $B = 180° - 42° - 45.5° = 92.5°$

🎯 ASA and AAS Case Solutions

ASA (Angle-Side-Angle)

The Situation: Two angles with the side BETWEEN them.

Quick Trick: Find the third angle first! (They add to 180°) Then use Law of Sines for everything else.

AAS (Angle-Angle-Side)

The Situation: Two angles and a side NOT between them.

Same method as ASA—find the third angle first!

Example (ASA)

Given: A = 50°, c = 10, B = 70°

Step 1: Find angle C $C = 180° - 50° - 70° = 60°$

Step 2: Use Law of Sines $\frac{a}{\sin A} = \frac{c}{\sin C}$ $a = \frac{10 \times \sin(50°)}{\sin(60°)} = \frac{10 \times 0.766}{0.866} ≈ 8.85$

$b = \frac{10 \times \sin(70°)}{\sin(60°)} = \frac{10 \times 0.940}{0.866} ≈ 10.85$

📐 Area Formulas: Five Powerful Methods!

Method 1: Area = ½ab sin C

The Classic Formula

When you know two sides and the angle between them: $\text{Area} = \frac{1}{2} \times a \times b \times \sin C$

Why does this work? Remember: Area = ½ × base × height. The height is one side times sine of the angle!

Example

Triangle with a = 6, b = 8, C = 30° $\text{Area} = \frac{1}{2} \times 6 \times 8 \times \sin(30°)$ $\text{Area} = \frac{1}{2} \times 6 \times 8 \times 0.5 = 12 \text{ sq units}$

Method 2: Area = rs (Semi-perimeter Formula)

What’s r? The inradius—radius of the circle that fits INSIDE the triangle, touching all three sides.

What’s s? The semi-perimeter = (a + b + c) / 2

$\text{Area} = r \times s$

Think of it: The inscribed circle “fills” the triangle, and this formula captures that relationship!

Example

Triangle with sides 5, 12, 13 (a right triangle!)

• s = (5 + 12 + 13) / 2 = 15
• Area (we know) = ½ × 5 × 12 = 30
• So: r = Area / s = 30 / 15 = 2

The inradius is 2 units!

Method 3: Area = abc / 4R (Circumradius Formula)

What’s R? The circumradius—radius of the circle that passes through ALL THREE vertices.

$\text{Area} = \frac{abc}{4R}$

Rearranged (to find R when you know the area): $R = \frac{abc}{4 \times \text{Area}}$

Example

Triangle with a = 3, b = 4, c = 5, Area = 6 $R = \frac{3 \times 4 \times 5}{4 \times 6} = \frac{60}{24} = 2.5$

The circumradius is 2.5 units!

🔴 The Inradius ® and Half-Angle Formulas

Inradius Formula

$r = (s-a)\tan\frac{A}{2} = (s-b)\tan\frac{B}{2} = (s-c)\tan\frac{C}{2}$

Also: $r = 4R \sin\frac{A}{2} \sin\frac{B}{2} \sin\frac{C}{2}$

Circumradius with Sine Formula

$R = \frac{a}{2\sin A} = \frac{b}{2\sin B} = \frac{c}{2\sin C}$

Example

For a triangle with A = 60°, a = 10: $R = \frac{10}{2\sin(60°)} = \frac{10}{2 \times 0.866} = \frac{10}{1.732} ≈ 5.77$

📊 Half-Angle Tangent Formulas

These formulas express tan(A/2), tan(B/2), tan(C/2) in terms of the sides!

The Key Formulas

$\tan\frac{A}{2} = \frac{r}{s-a} = \sqrt{\frac{(s-b)(s-c)}{s(s-a)}}$

$\tan\frac{B}{2} = \frac{r}{s-b} = \sqrt{\frac{(s-a)(s-c)}{s(s-b)}}$

$\tan\frac{C}{2} = \frac{r}{s-c} = \sqrt{\frac{(s-a)(s-b)}{s(s-c)}}$

Example

Triangle with sides a = 7, b = 8, c = 9

• s = (7 + 8 + 9) / 2 = 12
• s - a = 5, s - b = 4, s - c = 3

$\tan\frac{A}{2} = \sqrt{\frac{4 \times 3}{12 \times 5}} = \sqrt{\frac{12}{60}} = \sqrt{0.2} ≈ 0.447$

$\frac{A}{2} = \tan^{-1}(0.447) ≈ 24.1°$ $A ≈ 48.2°$

This matches our SSS solution earlier! 🎉

🎁 Quick Reference Summary

graph TD
    A["Know 3 Sides SSS"] --> B["Law of Cosines for Angles"]
    C["Know 2 Sides + Included Angle SAS"] --> D["Law of Cosines for 3rd Side"]
    D --> E["Law of Sines for Angles"]
    F["Know 2 Angles + 1 Side ASA/AAS"] --> G["Find 3rd Angle: A+B+C=180"]
    G --> H["Law of Sines for Sides"]

Area Formula Cheat Sheet

🚀 You Did It!

You now have a complete toolkit for:

• ✅ Solving ANY oblique triangle (SSS, SAS, ASA, AAS)
• ✅ Finding areas using multiple methods
• ✅ Working with inradius ® and circumradius ®
• ✅ Using half-angle tangent formulas

Remember: Every triangle problem is just a puzzle. Identify what you know, pick the right formula, and solve step by step. You’ve got this! 💪