A 45-45-90 triangle is a special right triangle with two angles measuring 45 degrees each and one right angle (90 degrees). This makes it an isosceles right triangle, meaning the two legs (sides opposite the 45-degree angles) are congruent. The sides of a 45-45-90 triangle have a specific ratio: if each leg has a length of $x$, then the hypotenuse (the side opposite the 90-degree angle) has a length of $x\sqrt{2}$. This relationship is key to solving problems involving these triangles.
Using this knowledge, you can quickly calculate missing side lengths given only one side! This is done by either multiplying or dividing by $\sqrt{2}$.
Match the term with its definition:
| Term | Definition |
|---|---|
| 1. Hypotenuse | A. The side opposite the right angle in a right triangle. |
| 2. Leg | B. Either of the two shorter sides of a right triangle that form the right angle. |
| 3. Isosceles Triangle | C. A triangle with two congruent sides. |
| 4. Right Triangle | D. A triangle with one angle measuring 90 degrees. |
| 5. $\sqrt{2}$ | E. The factor by which you multiply the leg to get the hypotenuse in a 45-45-90 triangle. |
(Answers: 1-A, 2-B, 3-C, 4-D, 5-E)
In a 45-45-90 triangle, the two ____ are always equal in length. If one leg measures 5 units, the other leg also measures ____ units. The hypotenuse will then measure ____ units.
(Answers: legs, 5, $5\sqrt{2}$)
Explain how knowing the properties of a 45-45-90 triangle can simplify solving geometric problems compared to using the Pythagorean theorem.
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