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Converse of the Pythagorean Theorem
If a^2 + b^2 = c^2, then the triangle is a right triangle, where a and b are the lengths of the legs and c is the length of the hypotenuse of the triangle.
Pythagorean Inequality Theorems
If c^2 > a^2 + b^2, then the triangle is obtuse and
Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
θ
A greek symbol used to represent an angle
sine of θ
sin(θ) =length of opposite side/length of hypotenuse
cosine of θ
cos(θ)=length of adjacent side/length of hypotenuse
tangent of θ
tan(θ)=length of opposite side/length of adjacent side
angle of depression
The angle below horizontal that an observer must look to see an object that is lower than the observer. Note: The angle of depression is congruent to the angle of elevation due to the alternate interior angles theorem
angle of elevation
The angle above horizontal that an observer must look to see an object that is higher than the observer. Note: The angle of elevation is congruent to the angle of depression due to the alternate interior angles theorem
Complementary Angles
The sum of the measure of two angles is 90 degrees.
Opposite side
In a right triangle, the side of the triangle opposite the vertex of one acute angle is called the opposite side, it is the leg not included to form the angle.
Similar triangles
Triangles are similar if they have the same shape but not necessarily the same size. Triangles whose corresponding angles are congruent are similar. Corresponding sides of similar triangles are all proportional.
For each pair of complementary angles in a right triangle,
the sine of one angle is equal to the cosine of its complement.
A right triangle
is a triangle that includes one right, or 90-degree, angle.
three sides
Like any triangle, a right triangle has
three angles
A triangle also has
Two of the sides
come together at the right angle
Hypotenuse
in a right triangle, the side opposite the right angle
Angle C
is always the right angle.
the angles of a Right Triangle
are labeled with upper-case letters A, B, and C.
P = a + b + c
The perimeter of a right triangle is equal to the sum of the lengths of the three sides. Using the conventional labels: In words: Perimeter (P) is equal to side a plus side b plus side c.
A = ½ab
The area of a right triangle is equal to one-half the product of side a and side b. Or in mathematical terms:
The Pythagorean Theorem
For any right triangle, the lengths of the two sides and the hypotenuse are related by an important mathematical statement called the
c^2 = a^2 + b^2
This formal statement of the Pythagorean Theorem relates the lengths of the three sides of a right triangle:
The hypotenuse
is always the longest of the three sides of a right triangle
The length of the hypotenuse
is always less than the sum of the two other sides
the length of the third side
Given the lengths of two sides, you can calculate
side c — the hypotenuse
If you are given the lengths of sides a and b, you can determine the length of
you can determine the length to the remaining side
If you are given the length of the hypotenuse and one other side (a or b)
a, b = the lengths of the other two sides
By solving the Pythagorean Theorem for c, you have an equation for determining the length of the hypotenuse: