45-45-90 Triangle: Leg a with Hypotenuse 2√6

The image displays a special right triangle, specifically a 45-45-90 isosceles right triangle. The diagram shows two legs of equal length $a$, two acute angles of $45^\circ$, and a hypotenuse labeled as $a\sqrt{2}$. The prompt indicates a scenario where the hypotenuse is given as $2\sqrt{6}$ and requires solving for $a$.

Answer

To find the value of $a$, we equate the given hypotenuse $2\sqrt{6}$ to the formulaic hypotenuse $a\sqrt{2}$. By dividing both sides by $\sqrt{2}$ and simplifying the radical, we find that $a = 2\sqrt{3}$.

Explanation

1. Identify the Triangle Properties In a $45^\circ-45^\circ-90^\circ$ triangle, the ratio of the sides is always constant. If the legs (the sides adjacent to the right angle) are $a$, then the hypotenuse is exactly $a$ multiplied by the square root of 2. $Legs = a, \quad Hypotenuse = a\sqrt{2}$ This relationship is derived from the Pythagorean Theorem where $a^2 + a^2 = c^2$.

2. Set up the Equation The problem states that the hypotenuse of a specific triangle is $2\sqrt{6}$. We match this numerical value to the algebraic expression for the hypotenuse provided in the diagram. $a\sqrt{2} = 2\sqrt{6}$ This equation represents the equivalence between the general rule for special triangles and the specific dimensions provided. ⚠️ This step is required on exams to demonstrate you understand the ratio of the sides.

3. Isolate the Variable $a$ To solve for $a$, divide both sides of the equation by $\sqrt{2}$. Use the property of radicals that allows you to divide terms under the same radical sign. $a = \frac{2\sqrt{6}}{\sqrt{2}}$ Dividing a square root by another square root allows the radicands to be divided directly.

4. Simplify the Radical Expression Simplify the fraction by dividing $6$ by $2$ within the radical. $a = 2\sqrt{\frac{6}{2}} = 2\sqrt{3}$ Since 3 is a prime number, the radical cannot be simplified further. ⚠️ This step is required on exams to ensure the answer is in simplest radical form.

Final Answer

The value of the side length $a$ is: $\boxed{2\sqrt{3}}$

Common Mistakes

• Incorrect Radical Division: Students often try to divide the coefficients by the radicand (e.g., trying to divide the $2$ outside the radical by the $\sqrt{2}$), which is algebraically invalid. You can only divide "outside numbers" by "outside numbers" and "inside numbers" by "inside numbers."
• Reciprocal Error: Some students mistakenly multiply the hypotenuse by $\sqrt{2}$ to find the leg ($a = 2\sqrt{6} \cdot \sqrt{2} = 4\sqrt{3}$). Remember: the hypotenuse is the longest side; to find a shorter leg, you must divide.