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Triangle is a polygon or simple closed figure formed by three-line segments or sides. It is composed of 3 sides or line segments, vertex points, and angles. Triangles are classified into six types, three types being based on their sides (Scalene Triangle, isosceles triangle, and Equilateral Triangle) and the other three types based on their angles (acute-angled triangle, right-angled triangle, and obtuse-angled triangle). Among the six types of triangles, the right-angled triangle is the one we are going to study further in detail.
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Key Takeaways: Triangle, right angle triangle, hypotenuse, construction of RHS, sides
Also read: Isosceles Triangle Theorems
What is a Right-Angled Triangle?
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A right-angled triangle has three sides. This type of triangle is often used in geometry which are referred to as the 'base', 'height', and 'hypotenuse', respectively. A right-angled triangle has one angle that is 90 degrees, while the other two are Acute angles. The longest side and the side inverse to the right angle of the right-angle triangle is known as 'hypotenuse’.
Properties of Right-Angled Triangle
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Following are the Properties of Right-Angled Triangle:
- One angle out of three is always a right one equal to 90°.
- The hypotenuse is always the side inverse to the angle equal to 90°.
- The longest side in a right-angled triangle is always the hypotenuse (H).
- The sum of the two interior angles of a right-angled triangle, other than the right one, is always 90 degrees.
- The other two sides of a right-angled triangle are known as the base(B) and perpendicular(P) that are adjacent to the right angle.
- The area of a right-angled triangle is equal to half of the sum of the right angle's two adjoining sides.
Also read: Quadrilateral Formula
Construction of RHS
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Geometry is one of the branches of mathematics dealing with shapes or figures along with their construction. There are a variety of shapes and figures in geometry and to draw or construct the same we need some tools. Following are the tools required to construct a right-angled triangle:
- Pencil
- Scale
- Compass
- Protractor
Step1: To begin, we shall draw a rough form of a triangle, named ABC.
Step 2: Define a line segment (BC) of length (8 cm).
Step 3: Draw a perpendicular line from point (B) to point (x) adjacent to the line (BC).
Step 4: With (C) being the center point, draw a curve that cuts the (BX) at the point (A) with the radius of (10 cm).
Step 5: Now, join the points (A to C).
As a result, (ABC) is the required right-angled triangle.
Points to Remember
Following are some important points:
- Right-angled triangle has 3 sides known as Hypotenuse (H), Base(B), and Perpendicular(P).
- The formula to calculate the area of a right-angled triangle is A = ½ (B × P).
- We get three equivalent triangles if we drop a perpendicular from the hypotenuse to the right angle.
- If a circumcircle is drawn across each of the three vertices, the radius of this respective circle is equal to half the length of the hypotenuse.
- If one angle is equal to 90° and the rest of the two angles of a right-angled triangle are equivalent to 45° each, then it is an Isosceles Right-Angled Triangle.
Also read: Properties of Determinants
Sample Questions
Ques: Find the area of the right-angled triangle, whose height is 7cm and base 8cm respectively. (2 Marks)
Ans: The given height and base are h = 7cm and b = 8cm.
Substituting the values of the base and height in the formula:
A = 1/2 bh
= 1/2 (8)(7)
= 28 cm2
Ques: Find out the area of a right-angled triangle with a hypotenuse of 15 cm and one side of 12 cm. (2 Marks)
Ans: AB2 = AC2 – BC2
= 152 – 122
= 225 – 144
= 81
Therefore, AB = 9
Hence, the area of the triangle = ½ × base × height
= ½ × 12 × 9
= 54 cm2
Ques: A right-angled triangle's hypotenuse is 16 units long, while one of the triangle's sides is 8 units long. Using the Pythagoras hypothesis formula, determine the percentage of the third side. (3 Marks)
Ans: 16 units = hypotenuse
Consider the provided side of a triangle as the height of a perpendicular = 8 units.
Substituting the given measurements into the Pythagoras hypothesis formula
Base2 + Height2 = Hypotenuse2
B2 + 82 = 162
B2 = 256 - 64
B = 192 = 13.856 units
As a result, the third side of a triangle is 13.856 units.
Ques: Construct a right triangle ABC with AC equaling 5 units and BC equaling 3 units. (3 Marks)
Ans: Follow the steps below to construct a right triangle ABC:
Step 1: Draw a point B on a horizontal line.
Step 2: Using 'B' as the center and estimating 3 units of width in a compass, draw two arcs on one or both sides of the line and name them as 'D' and 'C.'
Step 3: Using a compass and calculating 5 units as breadth taking 'D' as a center, draw a curve above 'B' and name it as 'A'.
Step 4: Recreate the interaction with 'C' as the center.
Step 5: Connect the points of intersection of these arc segments with 'B' and 'C'.
Step 6: Draw a 90° angle around point B.
Ques: The triangle's base and height are in the 3:2 proportion. If the triangle's area is 243 cm2, determine the triangle's base and height. (3 Marks)
Ans: Let the common ratio be x
Then the height of triangle = 2x
And the base of triangle = 3x
Area of triangle = 243 cm2
= 1/2 × b × h
⇒ 243 = 1/2 × 3x × 2x
⇒ 3x2 = 243
⇒ x2 = 243/3
⇒ x = √81
⇒ x = √ (9 × 9)
⇒ x = 9
Therefore, height of triangle = 2 × 9
= 18 cm
Base of triangle = 3x
= 3 × 9
= 27 cm
Ques: Calculate the area of a triangle with sides of 41 cm, 28 cm, and 15 cm. Take note of the length of the elevation about the triangle's longest side. (3 Marks)
Ans: Semi-perimeter of the triangle = (a + b + c)/2
= (41 + 28 + 15)/2
= 84/2
= 42 cm
Therefore, area of the triangle = √ (s (s – a) (s – b) (s – c))
= √ (42 (42 – 41) (42 – 28) (42 – 15)) cm2
= √ (42 × 1 × 27 × 14) cm2
= √ (3 × 3 × 3 × 3 × 2 × 2 × 7 × 7) cm2
= 3 × 3 × 2 × 7 cm2
= 126 cm2
Now, area of triangle = 1/2 × b × h
Therefore, h = 2A/b
= (2 × 126)/41
= 252/41
= 6.1 cm
Ques: Find out the area of a three-sided plot, whose sides are in the ratio of 2: 3: 4 with the perimeter of 180 m respectively. (3 Marks)
Ans: Let the common ratio be x, then the three sides of a triangle are 2x, 3x, 4x
Now, perimeter = 180 m
Therefore, 2x + 3x + 4x = 180
⇒ 9x = 180
⇒ x = 180/9
⇒ x = 20
Therefore, 2x = 2 × 20 = 40
3x = 3 × 20 = 60
4x = 4 × 20 = 80
Area of triangle = √ (s (s – a) (s – b) (s – c))
= √ (90(90 – 80) (90 – 60) (90 – 40))
= √ (90 × 10 × 30 × 50))
= √ (3 × 3 × 2 × 5 × 2 × 5 × 3 × 2 × 5 × 5 × 5 × 2)
= 3 × 2 × 5 × 2 × 5 √ (3 × 5)
= 300 √15 m2
= 300 × 3.872 m2
= 1161.600 m2
= 1161.6 m2
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