91Ó°ÊÓ

When we talk about the perimeter of any shape, we are referring to the total distance around the boundary of that shape. In the case of an isosceles right triangle, the perimeter is calculated by adding together the lengths of all three sides. Since two sides of this triangle are equal, being both legs, and the third side is the hypotenuse, our formula simplifies to:
- Perimeter = length of hypotenuse + 2 times the length of one leg.
For example, if the hypotenuse is 6 cm and each leg is 4.24 cm as found in our step-by-step solution, the perimeter would be calculated as 6 cm + 2 * 4.24 cm = 14.48 cm.
Thus, understanding how to calculate the perimeter helps us find the total boundary length of our isosceles right triangle quickly and efficiently.

The 45-45-90 triangle is a special type of right triangle where two angles are 45 degrees and one angle is 90 degrees. This particular configuration leads to specific side length relationships.
The two legs of this triangle are equal because the two angles opposite these legs are the same (each being 45 degrees). This symmetry simplifies many calculations and is what makes the 45-45-90 triangle so useful in geometry.
In any 45-45-90 triangle, the side ratios are consistent:

• The legs are equal in length.
• The hypotenuse is \( \rac{\sqrt{2}}{1} \) times as long as each leg.

This consistent relationship between the sides is a great tool to have when solving geometry problems quickly.

The relationship between the hypotenuse and the legs in a 45-45-90 triangle is key to solving problems involving this type of triangle. It is based on a fundamental geometric principle: In a 45-45-90 triangle, the hypotenuse is always the length of a leg multiplied by \( \sqrt{2} \).
To find the length of the legs when given the hypotenuse, you can rearrange the formula:

• Leg length = Hypotenuse ÷ \( \sqrt{2} \)

This formula allows us to find the other dimensions of the triangle when one side is known. For instance, with a hypotenuse of 6 cm, each leg will be \( \frac{6}{\sqrt{2}} = 4.24 \) cm, approximately.
Understanding this relationship helps in quickly determining unknown side lengths, making problem-solving faster and more intuitive.