The Triangle and its Properties

Right Angled Triangle

Have you seen the rooms in your house or in your schools? Are they not right-angled? Also in your schools, you must have noticed the badminton, basketball, volleyball courts. Aren’t they right-angled? Every one of us plays games like Chessboard, table tennis table,  ludo etc. All these are nothing but the examples of the right angle. Now further we are going to study the right angled triangle in detail.

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Right Angled Triangle

When we look at a right angle triangle we know that one angle of the triangle is always equal to 90°. A triangle where one of its interior angles is a right angle (90 degrees) the other two are acute angles. It is exactly a quarter of a circle.

Suppose when cutting a pizza into 4 equal parts, each part will form a right angle. The little square in the figure below, tell us that it is a right angle.  These triangles are used in many cases like in finding the distance of the slope if you only know its height.

The side opposite to the right angle is called as the hypotenuse while the other two sides are legs of the right angled triangle. The product of the two sides adjacent to the right angle is twice the area of the triangle.
Right Angled Triangle

Let this triangle be ΔABC and let ∠B = 90. So, in this case, the side opposite to the right angle is  AC and is called as the hypotenuse. Now, do you know the Pythagoras Theorem? Well, it states that the square of the hypotenuse is equal to the sum of the squares of the other sides. So by the Pythagoras property,

AC² =  AB² + BC²

Let us see an example: ΔABC is a right angled triangle at C. If AC = 5 cm, BC = 12 cm. Find the length of AB.

You can choose any of these sides as a base, but when working with right triangles it is usually quicker to use one of the right angle legs as a base, so the other right angle leg becomes the height

So we can say that, (AB)² = (AC)² + (BC)²  (using Pythagoras Theorem)
(AB)² = (5)² + (12)² =  25 + 144 = 169
AB = 13 cm

Solved Examples for You

Question 1. Any triangle with sides a, b, c, if a² + b² = c², then the angle between a and b measures

  1. 180°
  2. 100°
  3. 170°
  4. 90°

Answer : D. If a² + b² = c², c is the hypotenuse and the angle opposite to the hypotenuse is a right angled triangle in Pythagoras Theorem which lies between a and b and measure of right angled triangle is 90°. Therefore the correct answer is D

Question 2. A grassy land in the shape of a right angled triangle has its hypotenuse 11 meters more than twice the shortest side. If the third side is 77 metres more than the shortest side. The sides of the grassy land are:

  1. 8m, 17m, 15m,
  2. 2m, 16m, 13m,
  3. 10m, 4m, 5m,
  4. 7m, 10m, 14m,

Answer : A. Let the length of the shortest side be x meters. Then,

Hypotenuse = ( 2x + 1) meters, Third side = (x + 7) meters. The hypotenuse² = sum of the square of the remaining two sides

⇒ ( 2x + 1)² = x² + ( x + 7)²
⇒ 4x² + 4x  + 1 = 2x² +14x +49
⇒ 2x² – 10x – 48 = 0
⇒ x² – 5x – 24 = 0
⇒ x² – 8x + 3x – 24 = 0
⇒x (x – 8) + 3 ( x- 8) = 0
⇒x = 8, -3
So, ⇒x = 8

Hence the length of sides of the grassy land is 8m, 17m, 15m.

Question 3: What are the 3 angles of the right angle triangle?

Answer: The three interior angles in a right angle triangle are not pre-defined. However, one of the interior angles is 90 degrees (that is perpendicular to the base). Most importantly, all three angles always add up to 180 degrees. So, if one angle is 90 degrees then the other two will add to 90 degrees.

Question 4: What are the 3 4 5 triangles rule?

Answer: This rule is best known to determine with absolutely certainly that an angle is 90 degrees. In addition, as per this rule one side of the triangle measure, 3 and the adjacent side measure 4, and then the diagonal side between those two points must extent 5 in the directive for it to be a right triangle.

Question: 5: What does a right angle mean?

Answer: Simply, a right angle means an angle, which is exactly 90 degrees or perpendicular to the base, corresponding to a quarter turn.

Question 6: State the formula of a right-angle triangle and explain isosceles right-angled triangle?

Answer:  The area of a right angle triangle is ½ base × perpendicular.

An isosceles right-angled triangle refers to a triangle in which the adjacent sides to 90 degrees are equal in length to each other.

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