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Angle Measurement

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.

Degree:

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 (1') and th of a minute is one second (1'').
This is akin to hours division into minutes and seconds i.e., 1° = 60' and 1' = 60''

Radian:

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).

Relation between degree and radian:

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.

What is grade ?

Watching movie in a theater How can the geometric mean be used to watching movie in a theater ? When we are watching movie in a theater, we should sit at a distance that allows us to see all of the details in the movie. The distance that creates the best view is the geometric mean of the distance from the top of the theater screen to eye level and the distance from the bottom of the theater screen to eye level.
Geometric Mean

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: .

Ex 1: Find the geometric mean between each pair of numbers: (i) 16, 9 and (ii) 3, 12.
Sol: (i) Let 'x' be a geometric mean. (ii) Let 'x' be a geometric mean.
⇒ x2 = 144 ⇒ x2 = 36
⇒ x = √(144) ⇒ x = √(36)
⇒ x = 12 ⇒ x = 6
Ex 2: If is the geometric mean between 'a' and 2, then what is the value of 'a'?
Sol: Geometric mean between 'a' and 2 = ⇒ a = 9.

The relevance of geometric mean in right triangles is explained below.

Similarity in Right Triangles

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.

Pythagorean Theorem

The Pythagorean theorem is named after the Greek mathematician Pythagoras, which defines the relationship between the three sides of a right triangle. This theorem says that:
In a right-angled triangle, the sum of the squares of the measures of the legs (sides adjacent to the right angle) equals the square of the measure of the hypotenuse (side opposite to the right angle).



Converse of the Pythagorean theorem:
"If the sum of the squares of the measures of two sides of a triangle equals the square of the measure of the longest side, then the triangle is a right angled triangle". The converse of the Pythagorean theorem can be used to determine whether three measures of the sides of a triangle are those of a right triangle.

Pythagorean Triple

A set of three positive integers (a, b, c) that satisfy the equation a2 + b2 = c2, where 'c' is the greatest number, is called the Pythagorean triple. The smallest and best–known Pythagorean triple is (a, b, c) = (3, 4, 5), in which the sides of a right-angled triangle are in the ratio 3 : 4 : 5.

A primitive Pythagorean triple is a Pythagorean triple with no common factors except 1, ie., (Pythagorean triple with co-primes).
Ex: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), (20, 21, 29), (12, 35, 37), (9, 40, 41), etc.
The Pythagorean triple (3, 4, 5) is the only primitive Pythagorean triple involving consecutive positive integers.

If the measures of the sides of any right triangle are positive integers, then the measures form a Pythagorean triple.
Ex: 5, 12, 13; 11, 60, 61; 39, 80, 89; 28, 195, 197; 68, 285, 293, etc.

Note: If (a, b, c) is a Pythagorean triple, then (na, nb, nc) is also a Pythagorean triple for any positive integer 'n'.

Applications of Pythagorean theorem

Real Life Applications

Ex 1: Robert wants to instal a wall-mounted HDTV in his home theatre. The distance between the wall and sofa is 8 feet. If the TV manufacturer recommends a viewing distance of twice the diagonal of the TV screen, which of the following size suits him better?
a) 36" × 28" b) 38" × 30"
Sol: Using the Pythagorean theorem, we calculate the diagonal (D) of the TV screen.
a) D1 = ≈ 45.6".
Optimum viewing distance = 2 × 45.6 = 91.2" = = 7.60'.
b) D2 = ≈ 48.4".
Optimum viewing distance = 2 × 48.4 = 96.8" = ≈ 8.07'.
Since D2 is closer to the 8' distance between the TV and sofa in the room, option (b) suits better.
Practical example of Pythagorean theorem
Ex 2: In a game of baseball, you are at the first base and just picked a ground ball. You spot an opponent running toward third base already rd from the second base with an average speed of 15 ft/sec. At what speed should you throw the ball to run him out ?
Sol: Refer adjacent figure. The shortest distance from the first base to third base (D) is calculated from the Pythagorean theorem.
The shortest distance (D) = = 90√2 ≈ 127.28'
Balance distance to be covered by runner = 90 - 30 = 60'
Average speed of the runner = 15 ft/sec
Time taken by the runner = = 4 sec
So, to run him out, your throw should reach the third base in less than 4 sec.
Hence, the throw speed should be greater than or 31.82 ft/sec.
Other Applications of Pythagorean Theorem

Co-ordinate geometry:
Distance between two points (x1, y1) and (x2, y2) is calculated as:
d = .
Refer adjacent figure.

Vector algebra:
For the addition of two vectors.

Architecture and construction:
In construction of triangular shaped roofs, pyramids, stair-cases, ramps, etc.

Art designs etc.

You can take up a mini project on creating a beautiful Pythagorean spiral !

Special Right Triangles

A right triangle with some regular feature that makes calculations on triangle easier is called a special right triangle. There are two types of special right triangles: (i) angle-based special right triangles and (ii) side-based special right triangles.

A side-based special right triangle is one in which the lengths of the sides form a Pythagorean triple such as 3, 4 and 5. In this topic, we are discussing only about angle-based special right triangles.

The angle-based special right triangles are specified by the relationships of the angles of which the triangle is composed. The angles of these triangles are such that the larger angle, which is 90 degrees or π/2 radians, is equal to the sum of the other two angles. There are two types of angle-based special right triangles: (i) 45°–45°–90° triangle and (ii) 30°–60°–90° triangle.

  • 45°–45°–90° triangle: A right-angled triangle whose angle measures are 45°, 45° and 90° is called a 45°–45°–90° triangle. As it is an isosceles triangle, the two sides containing the right angle are equal, say 'x' units. Hence the length of the hypotenuse is given by √(x2 + x2) = (√2).x i.e., 'x' times the length of a leg. Therefore, in a 45°–45°–90° triangle, the ratio of the sides is 1 : 1 : √2.
  • 30°–60°–90° triangle: A right-angled triangle whose angle measures are 30°, 60° and 90° is called a 30°–60°–90° triangle. In this triangle, the length of the hypotenuse is twice the length of the shorter leg and the length of the longer leg is √3 times the length of the shorter leg. Therefore, in a 30°–60°–90° triangle, the ratio of the sides is 1 : √3 : 2.

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