30-60-90 Triangles Concept Explanation:
A 30-60-90 triangle is a special right triangle where the angles are always 30°, 60°, and 90°. This triangle has a fixed ratio between its side lengths, making it easier to solve for unknown sides once one side is known.
The side lengths follow the ratio:
- The side opposite the 30° angle is ( x ).
- The side opposite the 60° angle is ( x\sqrt{3} ).
- The hypotenuse (opposite the 90° angle) is ( 2x ).
This ratio helps us solve for missing sides without needing more advanced trigonometry.
Diagram:
Let’s draw a labeled diagram of a 30-60-90 triangle:
*** QuickLaTeX cannot compile formula:
\begin{array}{c}
\begin{tikzpicture}
\draw (0,0) -- (4,0) -- (4,3) -- cycle;
\draw (4,0) -- (4,3) node[midway, right] {x};
\draw (0,0) -- (4,3) node[midway, above] {x\sqrt{3}};
\draw (0,0) -- (4,0) node[midway, below] {2x};
\draw (0.5, 0.5) node {30^\circ};
\draw (3.5, 2.7) node {60^\circ};
\draw (4.2, 0.2) node {90^\circ};
\end{tikzpicture}
\end{array}
*** Error message:
Missing $ inserted.
leading text: \begin{array}{c}
Missing $ inserted.
leading text: ...0,0) -- (4,3) node[midway, above] {x\sqrt{
Extra }, or forgotten $.
leading text: ...) -- (4,3) node[midway, above] {x\sqrt{3}}
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leading text: \draw
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leading text: \end{tikzpicture}
Missing $ inserted.
leading text: \end{tikzpicture}
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leading text: \end{tikzpicture}
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leading text: \end{tikzpicture}
Extra }, or forgotten \endgroup.
leading text: \end{tikzpicture}
Extra }, or forgotten \endgroup.
leading text: \end{tikzpicture}
Missing } inserted.
leading text: \end{array}
Missing } inserted.
leading text: \end{array}
This diagram shows the relationship between the side lengths and angles in the triangle.
Hard Questions:
Q1: In a 30-60-90 triangle, the length of the side opposite the 30° angle is 6 units. Find the length of the hypotenuse and the side opposite the 60° angle.
Step-by-Step Solution:
Given:
- Side opposite 30°: ( x = 6 )
Using the fixed side ratios:
- Hypotenuse = ( 2x = 2(6) = 12 )
- Side opposite 60° = ( x\sqrt{3} = 6\sqrt{3} )
Answer:
- Hypotenuse = 12 units
- Side opposite 60° = ( 6\sqrt{3} ) units
Q2: The hypotenuse of a 30-60-90 triangle is 18 units. Find the length of the side opposite the 30° and 60° angles.
Step-by-Step Solution:
Given:
- Hypotenuse = 18
We know the hypotenuse is ( 2x ), so:
- ( x = \frac{18}{2} = 9 )
Now, using the side length ratios:
- Side opposite 30° = ( x = 9 )
- Side opposite 60° = ( x\sqrt{3} = 9\sqrt{3} )
Answer:
- Side opposite 30° = 9 units
- Side opposite 60° = ( 9\sqrt{3} ) units
Q3: In a 30-60-90 triangle, the side opposite the 60° angle is ( 9\sqrt{3} ) units. Find the lengths of the other two sides.
Step-by-Step Solution:
Given:
- Side opposite 60° = ( 9\sqrt{3} )
From the ratio, we know the side opposite the 60° angle is ( x\sqrt{3} ), so:
- ( x\sqrt{3} = 9\sqrt{3} ), therefore, ( x = 9 )
Now, using the ratios:
- Side opposite 30° = ( x = 9 )
- Hypotenuse = ( 2x = 18 )
Answer:
- Side opposite 30° = 9 units
- Hypotenuse = 18 units