Q: Which is the 30o 60o right triangle theorem?

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90

If its a 300-600-right angle triangle then the third angle must be 90 degrees. Then its base squared plus its height squared equals its hypotenuse squared usually written in the form of: a2+b2 = c2

Certainly not. In an equilateral triangle, in which all sides and all angles are congruent (or in other words, are the same) then all angles are 60o angles, there is no 90o angle and therefore it is not a right triangle. A typical right triangle would have angles of 90o, 60o, and 30o, and each of the three sides would be a different length.

The angles of a triangle always add up to 180o. A right angle triangle has angle of 90o, 60o, 60o.

I guess you are asking of the two sides which are not the hypotenuse, why the longer is root 3 times the shorter. Consider a 30, 60, 90 triangle ABC with ∠A = 30o, ∠B = 90o and ∠C = 60o and so side AC is the hypotenuse and BC is the shortest side. Reflect the triangle ABC in side BC to create triangle ABE. ∠EAB will be 30o, ∠ABE 90o and ∠BEA 60o. Side AE = AC. ∠EAC = ∠EAB + ∠BAC = 30o + 30o = 60o So triangle ACE is an equilateral triangle and thus side EC = AC Since EC = AC and EB = BC, EC=EB + BC ⇒ AC = BC + BC ⇒ AC = 2BC Using Pythagoras on triangle ABC to find length AB gives: AB2 + BC2 = AC2 ⇒ AB2 + BC2 = (2BC)2 ⇒ AB2 = 4BC2 - BC2 ⇒ AB2 = 3BC2 ⇒ AB = √3BC ie the shortest side is root 3 times the other non-hypotenuse side.

Related questions

90

This is a fun question. If the triangle rests on one of its sides as a base, then the altitude of the triangle, a line drawn from the apex of the triangle to the base, divides the triangle into two right angle 30o,60o, 90o, triangles. For convenience let each of the equal sides of the triangle 2 units. Then Pythagorus tells us that the base of each of these right triangles is 1 unit and the altitude is √3. This leads directly to sin(30o) cos(60o) 1/2 and cos(30o) sin(60o) √3/2 0.866 rounded. You can also use one of these right angle triangles to find the sine and cosine of 15o but the algebra gets a little messy.

If its a 300-600-right angle triangle then the third angle must be 90 degrees. Then its base squared plus its height squared equals its hypotenuse squared usually written in the form of: a2+b2 = c2

Certainly not. In an equilateral triangle, in which all sides and all angles are congruent (or in other words, are the same) then all angles are 60o angles, there is no 90o angle and therefore it is not a right triangle. A typical right triangle would have angles of 90o, 60o, and 30o, and each of the three sides would be a different length.

The angles of a triangle always add up to 180o. A right angle triangle has angle of 90o, 60o, 60o.

Sum of all angles in a triangle is equal to 180o. If two angles are 60o then the third angle is: Third angle = 180o - (60o + 60o) = 60o. But a right triangle is one having one angle equal to 90o but in this case there is no angle equal to 90o. So there is no right triangle which has two angles equal to 60o.

A regular triangle would have all angles equal, each being 60o. Thus a right angled triangle is an "irregular" triangle in that one of its angles is 90o which is not 60o.

180

I guess you are asking of the two sides which are not the hypotenuse, why the longer is root 3 times the shorter. Consider a 30, 60, 90 triangle ABC with ∠A = 30o, ∠B = 90o and ∠C = 60o and so side AC is the hypotenuse and BC is the shortest side. Reflect the triangle ABC in side BC to create triangle ABE. ∠EAB will be 30o, ∠ABE 90o and ∠BEA 60o. Side AE = AC. ∠EAC = ∠EAB + ∠BAC = 30o + 30o = 60o So triangle ACE is an equilateral triangle and thus side EC = AC Since EC = AC and EB = BC, EC=EB + BC ⇒ AC = BC + BC ⇒ AC = 2BC Using Pythagoras on triangle ABC to find length AB gives: AB2 + BC2 = AC2 ⇒ AB2 + BC2 = (2BC)2 ⇒ AB2 = 4BC2 - BC2 ⇒ AB2 = 3BC2 ⇒ AB = √3BC ie the shortest side is root 3 times the other non-hypotenuse side.

No, it is impossible to draw an equiangular right triangle. An equiangular triangle has three 60o angles. A right triangle has one 90o angle, and two 45o angles.

An equilateral triangle has 3 sides of equal length (that's what the word equilateral means) and also 3 equal angles, all of which are 60o. A right triangle, by definition, has one right angle of 90o. So if all the angles are 60o then it is not possible to have an angle of 90o, right? COOL!!!!!!!!!I GOT AN A+ on my portfolio thanks to this answer

Not possible in this Universe. Smallest regular polygon is a triangle when interior angles are 60o. If exterior angles are 30o the polygon has 360/30 ie 12 sides.