back to geometry
Properties of the Right Triangles
The pythagorean theorem
The square of the length of the hypotenuse of a right triangle is equal to the sum of the square of the lengths of the legs.
In right triangle ABC, AC
^{2}
= AB
^{2}
+ BC
^{2}
In right triangle ABC, there is one hypotenuse which opposite the right angle. There are two legs which are two sides of the right angle.
hypotenuse
^{2}
= Leg
_{1}
^{2}
+ Leg
_{2}
^{2}
Converse of the pythagorean theorem
In a triangle, if the square of the length of a side is equal to the sum of the square of the lengths of other two sides, then the triangle is a right triangle.
In the figure above, if AB
^{2}
= AC
^{2}
+ BC
^{2}
, then the triangle ABC is a right triangle.
or if c
^{2}
= a
^{2}
+ b
^{2}
, then the triangle ABC is a right triangle.
Note: in triangle ABC, a is the side BC which opposite the angle A, b is the side AC which opposite the angle B and c is the side AB which opposite the angle C.
Median throrem of a right triangle
In a right triangle, the length of the median of hypotenuse drawn from the hypotenuse to its opposite angle is one-half the length of the hypotenuse.
In the figure above, if triangle ABC is a right triangle, AB is the hypotenuse and D is the midpoint of AB, then
CD = (1/2)AB
CD is the median of the hypotenuse.
Property of the 30
^{o}
-60
^{o}
-90
^{o}
right triangle
In a right triangle, if an angle is 30 degrees, then the length of its opposite side is one-half the length of the hypotenuse.
In right triangle ABC, if angle B = 30
^{o}
, then AC = (1/2)AB
note: AB is the hypotenuse of the right triangle ABC. AC is a leg which opposite the angle B.
Corollary of the 30
^{o}
-60
^{o}
-90
^{o}
right triangle
In a right triangle, if the length of a leg is one-half the length of the hypotenuse, then the angle opposite this leg is 30 degrees.
In right triangle ABC, if AC = (1/2)AB, then
the angle B = 30
^{o}
note: AB is the hypotenuse and AC is a leg. The hypotenuse is opposite the right angle.
The 45
^{o}
-45
^{o}
-90
^{o}
right triangle
In a right triangle, if an angle is 45 degrees, then the length of the hypotenuse is the product of square root 2 and a leg.
In right triangle ABC, since angle A = 45
^{o}
So angle B = 90
^{o}
- 45
^{o}
= 45
^{o}
Since angle A = angle B = 45
^{o}
So AC = BC (In a triangle, equal angles opposite equal sides.)
In right triangle ABC, AB is the hypotenuse. AC and BC are legs.
AB
^{2}
= AC
^{2}
+ BC
^{2}
Since AC = BC, so
AB
^{2}
= 2AC
^{2}
The length of the hypotenuse is equal to the length of either leg multiplied by square root of 2.
The length of either leg is equal to one-half the length of the hypotenuse multiplied by square root of 2.
Definition of sine, cosine and tangent
Example 1:
In right triangle ABC, if the angle B = 30
^{o}
and AB = 6, what is the length of BC?
Solution:
In the right triangle ABC, since angle B = 30
^{o}
and angle B opposite AC,
and since angle C = 90
^{o}
and angle C opposite AB, so AB is the hypotenuse,
so AC = (1/2)AB = (1/2) × 6 = 3
(In a right triangle, the length of the leg opposite 30
^{o}
is one-half the length of the hypotenuse.)
AB
^{2}
= AC
^{2}
+ BC
^{2}
(the pythagorean theorem)
BC
^{2}
= AB
^{2}
- AC
^{2}
= 6
^{2}
- 3
^{2}
= 36 - 9 = 27 = 3
^{2}
× 3
BC = 3(square root of 3)
Example 2:
In the right triangle ABC, if the angle B = 45
^{o}
and BC = 5. Find the length of AB.
Solution:
In right triangle ABC, since angle B = 45
^{o}
and angle C = 90
^{o}
,
so angle A = 90
^{o}
- 45
^{o}
= 45
^{o}
so AC = BC = 5 (In a triangle, equal angles opposite equal sides)
then AB
^{2}
= AC
^{2}
+ BC
^{2}
= 2AC
^{2}
= 2 × 5
^{2}
so AB = 5(square root of 2)
Example 3:
In the triangle ABC, angle B = 45
^{o}
, angle C = 30
^{o}
and AD is perpendicular to BC at D. AD = 2, find the length of BC. (point D lies on BC)
Solution:
Since AD is perpendicular to BC at point D (Given), so
triangle ABD and triangle ACD are right triangles.
In right triangle ACD,
since angle C = 30
^{o}
and AD = 2,
so AC = 2 AD = 2 × 2 = 4
CD
^{2}
= AC
^{2}
- AD
^{2}
= 4
^{2}
- 2
^{2}
= 16 - 4 = 12 = 3 × 2
^{2}
CD = 2(square root of 3)
In right triangle ABD,
since angle B = 45
^{o}
, so angle BAD = 45
^{o}
so BD = AD = 2 (triangle ABD is an isosceles right triangle)
BC = BD + CD = 2 + 2(square root of 3) = 2(1 + square root of 3)