Įxample of a 45-45-90 Triangle EXAMPLE 3:įind the length of the hypotenuse of a 45-45-90 triangle whose legs are each in length. Then the hypotenuse can be found by the Pythagorean Theorem:,. Since the sides of a square are equal, we can label each leg of a 45-45-90 triangle as a. Such a triangle is called a 45-45-90 triangle, because half each of 90 o angle is a 45 o angle: Īnother special right triangle results when you cut a square in half using one of its diagonals. We do that by multiplying top and bottom by. We need to simplify this by rationalizing the denominator.
EXAMPLE 2:įind the exact length of the hypotenuse in this right triangle:Ĭomparing with a standard 30-60-90 triangle, the given side is, so, and. Thus, a = 4, and since side is adjacent the 30 o angle, its length is, so in this case. In this triangle, side is the hypotenuse, so we set its length to 2 a: 8 = 2 a. SOLUTION: "Exact" means we must give a simplified answer involving square roots. Notice that we can simplify by separating into the product of two square roots:Įxamples of 30-60-90 Triangles EXAMPLE 1:įind the exact length of the side marked x in the triangle on the right: Then the side opposite the 30 o angle will be of length a, and we can find the third side as follows: More specifically, let the length of the hypotenuse be 2 a. The other side can then be found by the Pythagorean Theorem, and will always be a multiple of the square root of 3. In general, in a 30-60-90 triangle, the side opposite the 30 o angle is half the hypotenuse. If a side of the equilateral triangle is 2 units in length, then the side opposite the 30 o angle in one of the 30-60-90 triangles is half that, or 1 unit, and the third side (the side opposite the 60 o angle) can be found by the Pythagorean Theorem: The acute angles of these triangles are 30 o and 60 o, so we call them "30-60-90 triangles": When an equilateral triangle is cut in half, you get two right triangles. Since the angles of any triangle add to 180 o, each angle of an equilateral triangle must be 60 o. From the Isosceles Triangle Theorem if follows that the angles of an equilateral triangle are all the same. Hints for Shorter Leg to Hypotenuse: MULTIPLY by 2 Hypotenuse to Shorter Leg: DIVIDE by 2 Shorter Leg to Longer Leg: MULTIPLY by Longer Leg to Shorter Leg: DIVIDE by You always want to work with the Shorter Leg…it makes it easier!ġ1 30-60-90 Shorter Leg Longer Leg = Shorter Hypotenuse = 2Shorter 30oġ2 Example 2: Find the values of x and y.ġ3 You Try: Find the value of each variable.An equilateral triangle is a triangle with all sides congruent. The shorter leg is across from the smaller angle (30o) The longer leg is across from the larger angle (60O)ĩ 30o Hypotenuse Longer Leg 60o Shorter Legġ0 You always want to work with the Shorter Leg…it makes it easier! We know that the hypotenuse is directly across from the 90O angle. In a triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is times as long as the shorter leg.Ĩ How do you know which is the shorter leg and which is the longer leg? Hypotenuse to Leg: DIVIDE by Hypotenuse = LEG LEG LEG In a triangle, the hypotenuse is times as long as each leg.ĥ Hints for 45-45-90 Leg to Hypotenuse: MULTIPLY by The sides of a right triangle satisfy this theorem: a2 + b2 = c2 Hypotenuse LEG Right triangles have one 90o angle The longest side is called the HYPOTENUSE It is directly across from the 90o The other sides are called LEGS Hypotenuse LEG Presentation on theme: "5.1 Special Right Triangles"- Presentation transcript:
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