30-60-90 Triangles - Varsity Tutors

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30-60-90 Triangles

Study Guide

Key Definition

A 30°−60°−90° triangle is a special type of right triangle whose sides are in the proportion $1 : \sqrt{3} : 2$.

Important Notes

  • The length of the hypotenuse is twice the length of the shorter leg.
  • The length of the longer leg is $\sqrt{3}$ times the length of the shorter leg.
  • These values make the triangle a right triangle due to the Pythagorean theorem.

Mathematical Notation

$x$ represents the length of the shorter leg.$2x$ represents the length of the hypotenuse.$x\sqrt{3}$ represents the length of the longer leg.Note: The square root symbol ($\sqrt{}$) indicates that we take the positive root.

Why It Works

The values satisfy the Pythagorean theorem: $x^2 + (x\sqrt{3})^2 = (2x)^2$.

Remember

In a 30°−60°−90° triangle, the sides are in the ratio $1 : \sqrt{3} : 2$.

Quick Reference

Side Ratio:$1 : \sqrt{3} : 2$

Understanding 30-60-90 Triangles

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Beginner Explanation

A 30°-60°-90° triangle is a special right triangle with sides in the ratio $1 : \sqrt{3} : 2$.Now showing Beginner level explanation.

Practice Problems

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1

Quick Quiz

Single Choice QuizBeginner

If the shorter leg of a 30°-60°-90° triangle is $x$, what is the length of the hypotenuse?

A$x$B$2x$C$x\sqrt{3}$D$3x$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.2

Real-World Problem

Question ExerciseIntermediate

Tree Shadow Scenario

A tree casts a shadow of length $x$. If the angle between the ground and the sunlight is 30°, how tall is the tree?Show AnswerClick to reveal the detailed solution for this question exercise.3

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

If the hypotenuse of a 30°-60°-90° triangle is of length $2x$, what is the area of the triangle?

Show AnswerClick to reveal the detailed explanation for this thinking exercise.4

Challenge Quiz

Single Choice QuizAdvanced

If the longer leg of a 30°-60°-90° triangle is of length $x\sqrt{3}$, what is the perimeter of the triangle?

A$3x$B$(3+\sqrt{3})x$C$5x$D$6x$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.

Recap

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