SAT-70: Pythagorean Theorem and the Distance Formula

SAT-70: Pythagorean Theorem and the Distance Formula

Description: The Pythagorean theorem relates the sides of a right triangle, and the distance formula is the very same idea used to measure between two points on a graph. Both appear constantly on the SAT.

The Pythagorean theorem

For a right triangle with legs a and b and hypotenuse c (the side opposite the right angle): a² + b² = c².

The hypotenuse is always the longest side and sits across from the 90° angle. (Oʻzbekcha: gipotenuza — toʻgʻri burchak qarshisidagi eng uzun tomon.)

Common Pythagorean triples (memorize these)

Whole-number right triangles save time: 3-4-5, 5-12-13, 8-15-17, and their multiples (like 6-8-10). If you spot two of the numbers, you know the third. (Oʻzbekcha: 3-4-5, 5-12-13 kabi butun sonli uchliklarni yodlab oling — vaqt tejaydi.)

The distance formula (same idea)

The distance between points (x₁, y₁) and (x₂, y₂) is:

d = √((x₂ − x₁)² + (y₂ − y₁)²)

It is just the Pythagorean theorem where the horizontal and vertical gaps are the two legs. (Oʻzbekcha: masofa formulasi — bu aslida Pifagor teoremasi; gorizontal va vertikal farqlar — katetlar.)

Worked Example 1 — find the hypotenuse

A right triangle has legs 6 and 8. Find the hypotenuse.

• c² = 6² + 8² = 36 + 64 = 100 → c = √100 = 10. (It's a 6-8-10 triple.)

Worked Example 2 — find a leg

A right triangle has hypotenuse 13 and one leg 5. Find the other leg.

• 5² + b² = 13² → 25 + b² = 169 → b² = 144 → b = 12.

(Oʻzbekcha: kateteni topish uchun gipotenuza kvadratidan ma'lum katet kvadratini ayiramiz.)

Worked Example 3 — distance on the plane

Find the distance between (1, 2) and (4, 6).

• Horizontal gap = 4 − 1 = 3; vertical gap = 6 − 2 = 4.
• d = √(3² + 4²) = √(9 + 16) = √25 = 5.

Tip: sketch the two points, make a right triangle from the horizontal and vertical gaps, then use a² + b² = c². (Oʻzbekcha: ikki nuqtadan toʻgʻri burchakli uchburchak yasab, Pifagordan foydalaning.)

The converse — checking for a right angle

The theorem also works backwards. If a triangle's sides satisfy a² + b² = c² (with c the longest side), then the triangle must have a right angle. So if the SAT gives sides 9, 40, 41, you can test 9² + 40² = 81 + 1600 = 1681 = 41² — yes, so it is a right triangle. This converse is a quick way to confirm a right angle without measuring. (Oʻzbekcha: agar a² + b² = c² boʻlsa, uchburchak toʻgʻri burchakli boʻladi — bu teskari teorema.)

Common mistake — which side is the hypotenuse

The formula only works if c is the hypotenuse (the longest side, opposite the right angle). If a problem gives you the hypotenuse and asks for a leg, you must subtract (leg² = c² − other leg²), not add. Adding when you should subtract is the most common error here, so always identify the hypotenuse first. (Oʻzbekcha: kateteni topishda qoʻshmaymiz, balki ayiramiz — avval gipotenuzani aniqlang.)

Practice 1

A right triangle has legs 9 and 12. Find the hypotenuse.

Practice 2

Find the distance between (−2, 1) and (1, 5).

Key words — Kalit soʻzlar

• Pythagorean theorem — Pifagor teoremasi
• Right triangle — toʻgʻri burchakli uchburchak
• Leg — katet
• Hypotenuse — gipotenuza
• Square / Square root — kvadrat / kvadrat ildiz
• Pythagorean triple — Pifagor uchligi
• Distance formula — masofa formulasi
• Coordinate plane — koordinata tekisligi
• Horizontal / Vertical — gorizontal / vertikal

Summary

• Right triangle: a² + b² = c²; the hypotenuse c is opposite the 90°.
• Know the triples 3-4-5, 5-12-13, 8-15-17 (and multiples) to save time.
• Distance formula is Pythagoras with legs (x₂ − x₁) and (y₂ − y₁).