SAT-71: Special Right Triangles — 45-45-90

SAT-71: Special Right Triangles — 45-45-90

Description: A 45-45-90 triangle always has the same side ratio, so you can find any side from one side instantly — no full Pythagorean calculation needed. It shows up in squares (cut along a diagonal) all over the SAT.

The fixed ratio

In a 45-45-90 triangle, the sides are in the ratio leg : leg : hypotenuse = 1 : 1 : √2.

The two legs are equal (it's isosceles, because the two 45° angles are equal), and the hypotenuse is a leg times √2. (Oʻzbekcha: 45-45-90 uchburchakda ikki katet teng, gipotenuza esa katet × √2 ga teng.)

Using the ratio both directions

• Leg → hypotenuse: multiply the leg by √2.
• Hypotenuse → leg: divide the hypotenuse by √2 (or multiply by √2 ÷ 2).

This triangle is exactly half of a square cut along its diagonal, so the diagonal of a square with side s is s√2. (Oʻzbekcha: kvadratning diagonali tomoni × √2 ga teng.)

Worked Example 1 — leg to hypotenuse

A 45-45-90 triangle has legs of 5. Find the hypotenuse.

• Hypotenuse = leg × √2 = 5√2.

Worked Example 2 — hypotenuse to leg

A 45-45-90 triangle has a hypotenuse of 10. Find each leg.

• Leg = hypotenuse ÷ √2 = 10 ÷ √2 = 10√2 ÷ 2 = 5√2.

(Oʻzbekcha: gipotenuzani √2 ga boʻlib, kateteni topamiz.)

Worked Example 3 — diagonal of a square

A square has side 7. Find the length of its diagonal.

• The diagonal splits the square into two 45-45-90 triangles with legs 7.
• Diagonal = 7√2 ≈ 9.9.

Tip: whenever you see a 45° angle or a square's diagonal, reach for the 1 : 1 : √2 ratio. (Oʻzbekcha: 45° burchak yoki kvadrat diagonali koʻrsangiz — 1 : 1 : √2 nisbatidan foydalaning.)

Worked Example 4 — area from the hypotenuse

A 45-45-90 triangle has a hypotenuse of 8√2. Find its area.

• First get a leg: leg = hypotenuse ÷ √2 = 8√2 ÷ √2 = 8.
• The two legs are perpendicular, so they are the base and height: area = ½ × 8 × 8 = 32.

(Oʻzbekcha: avval kateteni toping, soʻng yuza = ½ × katet × katet.)

Why the ratio works (quick proof)

Take a 45-45-90 triangle with each leg = 1. By the Pythagorean theorem the hypotenuse is √(1² + 1²) = √2. Scaling both legs to any length s scales the hypotenuse to s√2, which is exactly the 1 : 1 : √2 ratio. So the ratio is just the Pythagorean theorem applied once and remembered — that is why you never have to redo the square root. (Oʻzbekcha: nisbat — bu bir marta hisoblangan Pifagor teoremasi, shuning uchun uni har safar qayta hisoblamaymiz.)

Common mistake to avoid

Students sometimes multiply the hypotenuse by √2 to get a leg — that is backwards and makes the triangle too big. Remember the hypotenuse is the longest side, so going from hypotenuse to leg must make the number smaller (you divide by √2). (Oʻzbekcha: gipotenuza eng uzun tomon — undan kateteni topishda √2 ga boʻlinadi, koʻpaytirilmaydi.)

Practice 1

A 45-45-90 triangle has legs of 8. Find the hypotenuse.

Practice 2

The diagonal of a square is 6√2. Find the side length of the square.

Key words — Kalit soʻzlar

• 45-45-90 triangle — 45-45-90 uchburchak
• Special right triangle — maxsus toʻgʻri burchakli uchburchak
• Leg — katet
• Hypotenuse — gipotenuza
• Ratio — nisbat
• Isosceles — teng yonli
• Diagonal — diagonal
• Square — kvadrat
• √2 (radical) — √2 (ildiz)

Summary

• 45-45-90 sides are in ratio 1 : 1 : √2; the legs are equal.
• Leg → hypotenuse: ×√2; hypotenuse → leg: ÷√2.
• A square's diagonal = side × √2.