1) How can you draw a special 30º-60º-90º triangle from an equilateral triangle?
2) If you label the side lengths of the equilateral triangle 2, 2, and 2, how can this help you remember the ratios of side lengths in the 30º-60º-90º triangle?
2) If you label the side lengths of the equilateral triangle 2, 2, and 2, how can this help you remember the ratios of side lengths in the 30º-60º-90º triangle?
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1.
Draw an altitude from the vertex of the triangle. Since the triangle is equilateral the altitude will be a perpendicular bisector. The vertex will be bisected to form a 30° angle, perpendicular lines will form the 90° angle, and the other angle will be 60° because the angles of an equilateral triangle are equal.
2.
If each side is 2. Then the hypotenuse is simply a side which is 2. The side which is bisected by the perpendicular bisector is 2(1/2) = 1. Find the other side by Pythagorean theorem: 2² - 1² = x², x² = 3, so x = √3.
Hence the sides of a 30°-60°-90° triangle are in the ratio of 1:√3:2.
Draw an altitude from the vertex of the triangle. Since the triangle is equilateral the altitude will be a perpendicular bisector. The vertex will be bisected to form a 30° angle, perpendicular lines will form the 90° angle, and the other angle will be 60° because the angles of an equilateral triangle are equal.
2.
If each side is 2. Then the hypotenuse is simply a side which is 2. The side which is bisected by the perpendicular bisector is 2(1/2) = 1. Find the other side by Pythagorean theorem: 2² - 1² = x², x² = 3, so x = √3.
Hence the sides of a 30°-60°-90° triangle are in the ratio of 1:√3:2.
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It becomes much more apparent if you draw an actual triangle and bisect one of the angles. The hypotenuse would be a side, which is 2. And the side opposite the 30° angle would be the bisected side which is 2(1/2) = 1. Use a² + b² = c² to find the length of the altitude (side opposite 60°).
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