Question

special right triangles instruction active proving the 45°-45°-90° triangle theorem if the angles of a triangle are 45°, 45°, and 90°, show that the length of the hypotenuse is √2 times as long as each leg. 1. substitute the side lengths of the triangle into the pythagorean theorem. a² + a² = c² 2. combine like terms. a² = c²

Explanation:

Step1: Recall Pythagorean Theorem

The Pythagorean Theorem states that for a right triangle with legs of length \(a\) and \(b\) and hypotenuse of length \(c\), \(a^{2}+b^{2}=c^{2}\). In a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle, the two legs are equal (since the angles opposite them are equal, \(45^{\circ}\) each), so \(b = a\). Substitute \(b=a\) into the Pythagorean Theorem: \(a^{2}+a^{2}=c^{2}\).

Step2: Combine like terms

Simplify the left - hand side of the equation. \(a^{2}+a^{2}=(1 + 1)a^{2}=2a^{2}\), so we have \(2a^{2}=c^{2}\).

Step3: Solve for \(c\)

Take the square root of both sides of the equation \(2a^{2}=c^{2}\). Since \(c\) represents the length of a side of a triangle, it must be positive. So \(c=\sqrt{2a^{2}}=\sqrt{2}\cdot\sqrt{a^{2}}=\sqrt{2}a\) (because \(\sqrt{a^{2}} = a\) for \(a>0\), as \(a\) is a length). This shows that the length of the hypotenuse \(c\) is \(\sqrt{2}\) times the length of each leg \(a\).

Answer:

The hypotenuse \(c\) of a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle is \(\sqrt{2}\) times the length of each leg, proven by substituting \(b = a\) into the Pythagorean Theorem \(a^{2}+b^{2}=c^{2}\) (giving \(a^{2}+a^{2}=c^{2}\)), combining like terms to get \(2a^{2}=c^{2}\), and then taking the square root to find \(c=\sqrt{2}a\).