Question

consider triangle def. the legs have a length of 36 units each.
what is the length of the hypotenuse of the triangle?
18 units
18√2 units
36 units
36√2 units

Explanation:

Step1: Identify the triangle type

Triangle DEF is a right - isosceles triangle (since it has a right angle and two angles of \(45^{\circ}\), so the two legs are equal). For a right - triangle, we can use the Pythagorean theorem \(c^{2}=a^{2}+b^{2}\), where \(a\) and \(b\) are the legs and \(c\) is the hypotenuse. Here, \(a = b=36\).

Step2: Apply the Pythagorean theorem

Substitute \(a = 36\) and \(b = 36\) into the formula \(c^{2}=a^{2}+b^{2}\). So \(c^{2}=36^{2}+36^{2}\). Factor out \(36^{2}\): \(c^{2}=36^{2}(1 + 1)=36^{2}\times2\). Then take the square root of both sides: \(c=\sqrt{36^{2}\times2}\). Using the property of square roots \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (\(a\geq0,b\geq0\)), we get \(c = 36\sqrt{2}\).

We can also use the property of a \(45 - 45-90\) triangle, where the hypotenuse \(c\) is related to the leg \(l\) by the formula \(c=l\sqrt{2}\). Since \(l = 36\), then \(c = 36\sqrt{2}\).

Answer:

D. \(36\sqrt{2}\) units