45-45-90 Triangles Concept Explanation:
A 45-45-90 triangle is another special right triangle. It’s an isosceles triangle, meaning the two legs are equal in length. The angles in this triangle are always 45°, 45°, and 90°.
The side lengths have a fixed ratio:
- Both legs are ( x ).
- The hypotenuse is ( x\sqrt{2} ).
This ratio allows you to easily calculate missing sides once one side is known.
Diagram:
Here’s a diagram of a 45-45-90 triangle with labeled sides:
*** QuickLaTeX cannot compile formula:
\begin{array}{c}
\begin{tikzpicture}
\draw (0,0) -- (4,0) -- (4,4) -- cycle;
\draw (4,0) -- (4,4) node[midway, right] {x};
\draw (0,0) -- (4,4) node[midway, above] {x\sqrt{2}};
\draw (0,0) -- (4,0) node[midway, below] {x};
\draw (0.3, 0.3) node {45^\circ};
\draw (3.7, 3.7) node {45^\circ};
\draw (4.2, 0.2) node {90^\circ};
\end{tikzpicture}
\end{array}
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leading text: \begin{array}{c}
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leading text: ...0,0) -- (4,4) node[midway, above] {x\sqrt{
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leading text: ...) -- (4,4) node[midway, above] {x\sqrt{2}}
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leading text: \draw
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leading text: \end{tikzpicture}
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leading text: \end{tikzpicture}
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leading text: \end{tikzpicture}
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leading text: \end{tikzpicture}
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leading text: \end{tikzpicture}
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leading text: \end{tikzpicture}
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leading text: \end{array}
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leading text: \end{array}
Hard Questions:
Q1: In a 45-45-90 triangle, one leg is 5 units long. Find the length of the hypotenuse.
Step-by-Step Solution:
Given:
- One leg = 5
The hypotenuse in a 45-45-90 triangle is ( x\sqrt{2} ):
- Hypotenuse = ( 5\sqrt{2} )
Answer:
- Hypotenuse = ( 5\sqrt{2} ) units
Q2: The hypotenuse of a 45-45-90 triangle is ( 10\sqrt{2} ). Find the length of the legs.
Step-by-Step Solution:
Given:
- Hypotenuse = ( 10\sqrt{2} )
The hypotenuse is ( x\sqrt{2} ), so:
- ( x\sqrt{2} = 10\sqrt{2} ), therefore ( x = 10 )
Both legs are equal, so the legs are:
- Leg = 10
Answer:
- Legs = 10 units each
Q3: One leg of a 45-45-90 triangle is ( 7\sqrt{2} ) units long. Find the hypotenuse.
Step-by-Step Solution:
Given:
- One leg = ( 7\sqrt{2} )
The hypotenuse is ( x\sqrt{2} ), but the leg itself is already ( 7\sqrt{2} ), so:
- Hypotenuse = ( 7\sqrt{2} \times \sqrt{2} = 7 \times 2 = 14 )
Answer:
- Hypotenuse = 14 units