SAT-72: Special Right Triangles — 30-60-90

SAT-72: Special Right Triangles — 30-60-90

Description: Like the 45-45-90 of SAT-71, the 30-60-90 triangle always has the same side ratio, so one side gives you all three with no square-root work. It hides inside equilateral triangles cut in half, so it appears all over SAT geometry.

The fixed ratio

In a 30-60-90 triangle the sides are in ratio short leg : long leg : hypotenuse = 1 : √3 : 2.

The key is to match each side to the angle opposite it:

• The side opposite the 30° angle is the short leg (the "1").
• The side opposite the 60° angle is the long leg (the "√3").
• The side opposite the 90° angle is the hypotenuse (the "2"), always the longest.

(Oʻzbekcha: 30° qarshisidagi tomon eng kichik (1), 60° qarshisida √3, 90° qarshisida 2 (gipotenuza).)

How to use the ratio

Find the short leg first — everything is measured from it. The long leg is the short leg × √3, and the hypotenuse is the short leg × 2. If you are given a different side, work back to the short leg first. (Oʻzbekcha: avval kichik kateteni toping — qolgan tomonlar undan kelib chiqadi.)

Worked Example 1 — from the short leg

The short leg of a 30-60-90 triangle is 5. Find the other two sides.

• Long leg = 5 × √3 = 5√3.
• Hypotenuse = 5 × 2 = 10.

Worked Example 2 — from the hypotenuse

The hypotenuse is 14. Find the legs.

• Hypotenuse = short leg × 2, so short leg = 14 ÷ 2 = 7.
• Long leg = 7 × √3 = 7√3.

(Oʻzbekcha: gipotenuzani 2 ga boʻlib kichik kateteni topamiz.)

Worked Example 3 — from the long leg

The long leg is 6√3. Find the short leg and hypotenuse.

• Long leg = short leg × √3, so short leg = 6√3 ÷ √3 = 6.
• Hypotenuse = 6 × 2 = 12.

Where it comes from

Cut an equilateral triangle (all 60°) straight down the middle. You get two 30-60-90 triangles: the base is halved (short leg), the slanted side stays whole (hypotenuse, twice the short leg), and the height is the long leg. That is why the hypotenuse is exactly double the short leg. (Oʻzbekcha: teng tomonli uchburchakni teng ikkiga boʻlsak, 30-60-90 hosil boʻladi — shuning uchun gipotenuza kichik katetdan ikki barobar.)

Common mistake: putting √3 on the wrong leg. The √3 always goes with the 60° side, not the 30° side. (Oʻzbekcha: √3 doim 60° tomonida boʻladi.)

Worked Example 4 — area of an equilateral triangle

Find the area of an equilateral triangle with side 8.

• Drop a height to split it into two 30-60-90 triangles; the short leg is half the base = 4, so the height (long leg) = 4√3.
• Area = ½ × base × height = ½ × 8 × 4√3 = 16√3 ≈ 27.7.

(Oʻzbekcha: balandlik 30-60-90 dan kelib chiqadi, soʻng yuza = ½ × asos × balandlik.)

How to recognize a 30-60-90 on the test

The SAT rarely labels it for you. Look for these signals: a right triangle with a 30° or 60° angle marked; an equilateral triangle that has been split by a height; or a side that involves √3. Any one of those means the 1 : √3 : 2 ratio is waiting to save you the full Pythagorean calculation. (Oʻzbekcha: 30° yoki 60° burchak, yoki √3 li tomon koʻrsangiz — bu 30-60-90 uchburchak.)

Practice 1

The short leg of a 30-60-90 triangle is 9. Find the hypotenuse and long leg.

Practice 2

An equilateral triangle has side 10. Find its height.

Key words — Kalit soʻzlar

• 30-60-90 triangle — 30-60-90 uchburchak
• Short leg — kichik katet
• Long leg — katta katet
• Hypotenuse — gipotenuza
• Ratio — nisbat
• Opposite (angle) — qarshisidagi (burchak)
• Equilateral — teng tomonli
• Height (altitude) — balandlik
• √3 (radical) — √3 (ildiz)

Summary

• 30-60-90 sides are in ratio 1 : √3 : 2 (short : long : hypotenuse).
• Match each side to the angle opposite it; the √3 goes with the 60° side.
• Find the short leg first; long leg = ×√3, hypotenuse = ×2.