CBSE 10
CBSE 10 Geometry: Unravel the Enigma of Triangles (Solved Examples)
May 7, 2024
Aha
Language
English
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Triangles are one of the most important shapes in mathematics. They are widely used in geometry, trigonometry, and other math fields. Triangles are a three-sided polygon and are defined as a closed figure with three line segments as its sides.
A triangle is a closed figure with three line segments as its sides.
Triangles can have a variety of shapes. They can have one angle that is greater than 90 degrees, two angles that are greater than 90 degrees, or all angles that are less than 90 degrees. Triangles can also be classified based on the length of their sides.
The sum of the angles of a triangle is always 180 degrees. This is known as the Triangle Sum Theorem. The three angles of a triangle can be classified as acute, obtuse, or right.
There are many properties of triangles that are important to know. Some of these include the Triangle Inequality Theorem, which states that the sum of any two sides of a triangle must be greater than the third side, and the Law of Cosines, which is used to find the length of a side of a triangle when the lengths of the other two sides and the included angle are known.
There are several types of triangles, including scalene, isosceles, and equilateral triangles.
A scalene triangle is a triangle where all three sides have different lengths.
An isosceles triangle is a triangle where two sides have the same length.
An equilateral triangle is a triangle where all three sides have the same length.
An acute angled triangle is a triangle where all angles are less than 90 degrees.
A right angled triangle is a triangle where one angle is exactly 90 degrees.
An obtuse angled triangle is a triangle where one angle is greater than 90 degrees.
The perimeter of a triangle is the sum of the lengths of its sides.
The area of a triangle can be calculated using different formulas, depending on the information that is known about the triangle. One common formula is the base-height formula, which uses the length of the base and the height of the triangle. Another formula is Heron's formula, which uses the lengths of the sides to calculate the area.
The base-height formula for the area of a triangle is:
Area = (1/2) * base * height
where base is the length of the base of the triangle and height is the height of the triangle. The height of a triangle is the distance from the base to the opposite vertex.
Heron's Formula for the area of a triangle is:
Area = √[s(s - a)(s -b)(s - c)]
where a, b, and c are the lengths of the sides of the triangle, and s is the semi-perimeter of the triangle, which is half the sum of the lengths of the sides:
s = (a + b + c) / 2.
Heron's formula is useful when the lengths of the sides of the triangle are known, but the height is not.
Find the area of a triangle with base 6 cm and height 8 cm.
✅ Solution: Use the base-height formula: Area = (1/2) * base * height = (1/2) * 6 * 8 =
24 cm².
Find the area of an equilateral triangle with side length 6 cm.
✅ Solution:
1. The formula for the area of an equilateral triangle is:
Area = (√3/4)a2
where a is the length of the side.
2. Substituting a = 6, Area = (√3/4) * 62 = (√3/4) * 36 = 9√3 cm².
Find the area of a triangle whose sides are 5 cm, 12 cm, and 13 cm.
✅ Solution:
1. First, calculate the semi-perimeter of the triangle: s = (5 + 12 + 13) / 2 = 15.
2. Next, use Heron's Formula : Area = √[s(s - a)(s -b)(s - c)] =
√[15(15 - 5)(15 -12)(15 -13)] = √[15(10)(3)(2)] = √900 =
30 cm².
Triangles are a three-sided polygon and are defined as a closed figure with three line segments as its sides.
There are several types of triangles, including scalene, isosceles, and equilateral triangles.
Some of the properties of triangles include the Triangle Inequality Theorem, which states that the sum of any two sides of a triangle must be greater than the third side, and the Law of Cosines, which is used to find the length of a side of a triangle when the lengths of the other two sides and the included angle are known.
The perimeter of a triangle is the sum of the lengths of its sides, while the area of a triangle is the amount of space inside the triangle.
The formula for the perimeter of a triangle is the sum of the lengths of its sides, while the formula for the area of a triangle depends on the information that is known about the triangle.
A scalene triangle is a triangle where all three sides have different lengths. An isosceles triangle is a triangle where two sides have the same length. An equilateral triangle is a triangle where all three sides have the same length.
An acute angled triangle is a triangle where all angles are less than 90 degrees. A right angled triangle is a triangle where one angle is exactly 90 degrees. An obtuse angled triangle is a triangle where one angle is greater than 90 degrees.
In conclusion, triangles are an important shape in mathematics that have many different properties and applications. By understanding the different types of triangles and their properties, we can better understand and solve problems in geometry and other math fields. If you want to learn more about math and other subjects, check out Aha AI, a website that offers a wide range of educational resources.
