🎯 Learning Objectives
- Triangle types and their special properties.
- The Pythagorean theorem and the most famous Pythagorean triples.
- The two SAT special right triangles (45-45-90 and 30-60-90).
- Similarity and congruence: what they tell you and how to use ratios.
- 10 SAT-style practice questions.
1. Types of Triangles
Classified by Sides
Type | Sides | Angles |
Equilateral | All 3 equal | All 60° |
Isosceles | Exactly 2 equal | 2 equal angles opposite equal sides |
Scalene | All 3 different | All 3 angles different |
Classified by Angles
Type | Angle Property |
Acute | All angles < 90° |
Right | Exactly one 90° angle |
Obtuse | Exactly one angle > 90° |
2. The Triangle Inequality
Side c lies strictly between |a−b| and a+b
|a − b| < c < a + b
Any side of a triangle must be longer than the difference of the other two sides and shorter than their sum. If your numbers don't satisfy this, no triangle exists.
3. The Pythagorean Theorem
a, b are legs; c is the hypotenuse (longest side, opposite the right angle)
a² + b² = c²
Pythagorean Triples: Memorize These
Triple | Multiples Worth Knowing |
3-4-5 | 6-8-10, 9-12-15, 12-16-20 |
5-12-13 | 10-24-26, 15-36-39 |
7-24-25 | 14-48-50 |
8-15-17 | 16-30-34 |
⚡ Strategy: Recognize the Triple Instantly
A familiar triple can help when the given lengths and their roles match it. Check which side is the hypotenuse and verify a² + b² = c²; two familiar numbers alone do not determine the third side without those roles.
4. The Two Special Right Triangles
45° – 45° – 90° (Half a Square)
Hypotenuse = leg × √2
Sides ratio: 1: 1: √2
30° – 60° – 90° (Half an Equilateral)
Short leg: long leg: hypotenuse
Sides ratio: 1: √3: 2
💡 Pro Tip: Identify First, Compute Last
Whenever you see 45° or 30°-60° angles in a right triangle, immediately switch to the special-triangle ratios. Keep exact values, including square roots, when the question asks for them.
5. Similarity and Congruence
Two triangles are SIMILAR if they have the same shape (corresponding angles equal). Their sides are in proportion.
Two triangles are CONGRUENT if they have the same shape AND the same size (corresponding sides also equal).
Similar triangles share a ratio
If △ABC ~ △DEF, then AB/DE = BC/EF = AC/DF
❓ Same angles AND same side lengths? | ||
✅ YES ▼ | ❌ Same angles only ▼ | |
Congruent triangles | Similar triangles (sides in ratio) | |
6. Area of a Triangle
Area = (1/2) × base × height
The height must be perpendicular to the base. For right triangles, the two legs are perpendicular, so they serve as base and height directly.
7. Strategies
⚡ Memorize 3-4-5, 5-12-13, 7-24-25, 8-15-17
Spotting these saves you the Pythagorean theorem.
⚡ Spot 45° → ratio 1:1:√2
Hypotenuse = leg × √2.
⚡ Spot 30°-60° → ratio 1:√3:2
Short leg is half the hypotenuse.
⚡ For similar triangles, set up one proportion
Match corresponding sides; cross-multiply.
⚡ For an equilateral triangle of side s, area = s²·√3 / 4
Saves time on standard SAT figures.
8. Summary
📌 Key Takeaways
- Angle sum: 180° in every triangle.
- Pythagorean: a² + b² = c² (right triangles only).
- Triples: 3-4-5, 5-12-13, 7-24-25, 8-15-17.
- 45-45-90: ratio 1: 1: √2.
- 30-60-90: ratio 1: √3: 2.
- Similar triangles: matching angles, proportional sides.
- Area: (1/2) × base × height.
9. Practice: 10 Questions
Q1. A right triangle has legs 3 and 4. Its hypotenuse?
A) 5 B) 6 C) 7 D) √7
Hint: 3-4-5 triple.
Q2. A right triangle has legs 5 and 12. Hypotenuse?
A) 12 B) 13 C) 17 D) 25
Hint: 5-12-13 triple.
Q3. A 45-45-90 triangle has legs of length 7. Hypotenuse?
A) 7 B) 7√2 C) 14 D) √14
Hint: Hypotenuse = leg × √2.
Q4. A 30-60-90 triangle has hypotenuse 10. The shorter leg?
A) 5 B) 5√3 C) 10√3 D) 20
Hint: Short leg = hyp / 2.
Q5. A triangle has angles 50° and 60°. What is the third angle?
A) 60° B) 70° C) 80° D) 110°
Hint: 180 − 50 − 60 = 70.
Q6. The sides of a triangle are 6, 8, and x. Which is impossible?
A) 5 B) 8 C) 14 D) 15
Hint: Need x < 14, so 15 is invalid.
Q7. Two similar triangles have a side ratio of 2:3. The ratio of their areas?
A) 2:3 B) 4:9 C) 8:27 D) 1:1
Hint: Area scales by k².
Q8. An equilateral triangle has side 6. What is its area?
A) 6√3 B) 9√3 C) 12√3 D) 36
Hint: (s²√3)/4 = 9√3.
Q9. A right triangle has legs 7 and 24. Hypotenuse?
A) 23 B) 25 C) 27 D) 31
Hint: 7-24-25 triple.
Q10. In a 30-60-90 triangle, the short leg is 4. What is the long leg?
A) 4 B) 4√2 C) 4√3 D) 8
Hint: Long leg = short × √3.
Answer Key & Worked Solutions
# | Answer | Reasoning |
Q1. | A) 5 | 3-4-5 triple. |
Q2. | B) 13 | 5-12-13 triple. |
Q3. | B) 7√2 | Leg × √2. |
Q4. | A) 5 | Hyp / 2. |
Q5. | B) 70° | 180 − 110. |
Q6. | D) 15 | Triangle inequality fails. |
Q7. | B) 4:9 | Areas scale by k². |
Q8. | B) 9√3 | (36 × √3)/4. |
Q9. | B) 25 | 7-24-25 triple. |
Q10. | C) 4√3 | Long leg = short × √3. |