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Commonly used units of measurement of an angle are the degree and radian. These are relevant to any triangle and more so to a right-angled triangle.
If the rotation is th of one complete revolution, the measure is called one degree. It is denoted as 1°. th of a degree is one minute which is denoted as 1'. th of a minute is one second which is denoted as 1''. i.e, 1° = 60' and 1' = 60''Note that this is akin to division of hours into minutes and seconds.
The angle subtended at the center of a unit circle by an arc of unit length is defined as a radian (abbreviated as rad). Refer Fig (i). The symbol used to denote radians is c (i.e, c as an exponent).
We know that the circumference of a circle is 2πr. For a unit circle (r = 1 unit), the circumference is 2π. An arc of length 5 units, subtends an angle of 5 radians and that of 10 units subtends 10 radians. Therefore, one complete revolution (arc length becomes the circumference which is equal to 2π) subtends an angle of 2π radians.
In general, in a circle of radius r, an arc of length l will subtend an angle (θ) of () radians. i.e., or l = rθ Refer Fig (ii).
360° = 2π radians or 180° = π radians. Since π = , we have: 1° = radians = 0.0175 radians approx. and 1 radian = = 57°16' approx.
Note: Large angles (multiples of 360° or 2π radians) are measured as revolutions per second (rps) or revolutions per minute (rpm). Ex: The speed of a spinning wheel or a rotating shaft.
The geometric mean between two numbers is the positive square root of their product.
If a positive number 'x' is a geometric mean between two positive numbers 'p' and 'q',
then x = or x2 = pq.
This can be written using fractions as: .
The relevance of geometric mean in right triangles is explained below.
The altitude of a triangle corresponding to any side is the perpendicular segment from the opposite vertex to that side.
Consider a right triangle ABC. Draw the altitude BD
from the right angle B to the hypotenuse AC.
The altitude BD separates the
triangle ABC into two triangles called Δ ADB and Δ BDC. Compare the angles of three triangles by placing the angles on top of another.
The angles ∠4 and ∠7 have same measure as ∠1, the angles ∠6 and ∠8 have same measure as ∠2 and the angles
∠5 and ∠9 have same measure as ∠3. Therefore, by the AA (Angle–Angle) axiom of similarity, the two triangles Δ
ADB and Δ BDC are similar to the Δ ABC and to each other, that is, Δ ABC ∼ Δ ADB ∼ Δ BDC.
Since Δ ADB ∼ Δ BDC, the corresponding sides are proportional.
Thus, ⇒ BD2 = (AD)(CD).
Therefore, the measure of an altitude drawn from the vertex of the right angle to its hypotenuse is the geometric
mean between the measures of the two segments of the hypotenuse.
It is one of the methods to generate Pythagorean Triples.
Pythagoras and Latos formulas, Fibonacci's methods also generate Pythagorean Triples.
Let m and n be an arbitrary pair of integers
such that m > n > 0.
Let a = m2 – n2
b = 2mn
c = m2 + n2(It is obvious that a, b and c are also integers).Then the integers a, b, and c from a Pythagorean Triple.Ex: Let m = 5, n = 2a = m2 – n2 = 25 – 4 = 21b = 2mn = 2 × 5 × 2 = 20c = m2 + n2 = 25 + 4 = 29So (20, 21, 29) form a Pythagorean Triple.
Note: For a Pythagorean Triple,
(i) The product of the two sides containing the right angle is always divisible by 12.
(ii) The product of all the three sides of the triangle is always divisible by 60.