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Find Side Length x in 45-45-90 Right Triangle (Hypotenuse = 2)
Mathematics
Grade 10 (Junior High School)
Question Content
Find the length of side $x$ in simplest radical form with a rational denominator. The figure is a right triangle with a right angle, two 45-degree angles, hypotenuse length 2, and side $x$ opposite the 45-degree angle.
Correct Answer
$\sqrt{2}$
Detailed Solution Steps
1
Step 1: Identify the type of triangle. The triangle has one right angle and two 45° angles, so it is an isosceles right triangle (45-45-90 special right triangle).
2
Step 2: Recall the side ratio for a 45-45-90 triangle: the legs are congruent, and the hypotenuse is $\text{leg length} \times \sqrt{2}$. Let the leg length (which is $x$) be $a$. The formula is $\text{hypotenuse} = a\sqrt{2}$.
3
Step 3: Substitute the given hypotenuse length (2) into the formula: $2 = x\sqrt{2}$.
4
Step 4: Solve for $x$ by isolating it. Divide both sides by $\sqrt{2}$: $x = \frac{2}{\sqrt{2}}$.
5
Step 5: Rationalize the denominator by multiplying the numerator and denominator by $\sqrt{2}$: $x = \frac{2\sqrt{2}}{\sqrt{2} \times \sqrt{2}} = \frac{2\sqrt{2}}{2} = \sqrt{2}$.
Knowledge Points Involved
1
45-45-90 Special Right Triangle Properties
A 45-45-90 triangle is an isosceles right triangle, meaning its two legs are equal in length. The ratio of the sides is $1:1:\sqrt{2}$, where the first two values represent the leg lengths, and the third represents the hypotenuse length. This ratio allows quick calculation of unknown side lengths if one side is known.
2
Rationalizing the Denominator
Rationalizing the denominator is a process to eliminate radical expressions from the denominator of a fraction. For a denominator with a single square root $\sqrt{n}$, multiply both the numerator and denominator by $\sqrt{n}$ to convert the denominator to a rational number. This is required to present answers in standard "simplest radical form".
3
Triangle Angle Sum Theorem
The sum of the interior angles of any triangle is 180°. In this problem, this confirms that the two non-right angles must each be 45°, since $180° - 90° = 90°$, and the remaining two angles are equal, so $90° \div 2 = 45°$.
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