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An Introduction to Acute Angled Triangles
An acute angled triangle is a triangle that has all of its angles measuring less than 90 degrees. The three angles of an acute angled triangle always add up to less than 180 degrees. Acute angled triangles are always interesting to study because of their unique properties and the different shapes that they can create.
Classification of Triangles on the Basis of Their Sides is as Follows
The three types of triangles are equilateral, isosceles, and scalene.
Equilateral triangles have three sides of the same length.
Isosceles triangles have two sides of the same length.
Scalene triangles have three sides of different lengths.
Classification of Triangles on the Basis of Their Angles is as Follows
There are three types of triangles on the basis of their angles:
1. Equilateral Triangle:
An equilateral triangle has three equal angles of 60 degrees each. All sides of an equilateral triangle are of the same length.
2. Isosceles Triangle:
An isosceles triangle has two equal angles. The two shorter sides of an isosceles triangle are of the same length.
3. Scalene Triangle:
A scalene triangle has no equal angles. Each side is of a different length.
Definition of Acute Angled Triangle
A triangle with one angle measuring more than 90 degrees and two angles measuring less than 90 degrees.
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Acute Triangle Formulas
The acute triangle formulas are used to calculate the lengths of the sides and angles of an acute triangle.
The length of a side of an acute triangle can be calculated using the following formula:
s = √(b² + c² – a²)
The length of an angle of an acute triangle can be calculated using the following formula:
α = √(b² + c² – a²) / 2
The length of the hypotenuse of an acute triangle can be calculated using the following formula:
h = √(b² + c²)
Identification of the Acute Angled Triangle
The acute angled triangle is a geometric figure with three sides and three angles. The angles are all acute angles, meaning that they are less than 90 degrees. The sides are all straight lines and the angles are all congruent, meaning that they are all the same size.
Area of an Acute Angled Triangle
The area of an acute angled triangle can be found by using the Pythagorean theorem. The length of the base of the triangle, b, and the length of the height of the triangle, h, are used in the equation. The length of the triangle’s hypotenuse, c, is also used.
Perimeter of an Acute Angled Triangle
The perimeter of an acute angled triangle is the sum of the lengths of its sides.
Method to Check if a Given Triangle is Acute Angled or Not
The following is a method to check if a given triangle is acute angled or not:
1. Draw the triangle.
2. Measure the length of the triangle’s sides.
3. Calculate the angles of the triangle by using the following equation:
Angle = (180 – sum of the angles) /
4. Compare the calculated angle to the measure of the triangle’s angles. If the angle is less than the measure of the triangle’s angles, then the triangle is acute angled.
Properties of Acute Triangle
In an acute triangle, all three angles are less than 90 degrees. The side opposite the smallest angle is the shortest side, and the other two sides are the longest sides.
Solved Examples
For example, say we have a triangle ABC with sides measuring 5 cm, 6 cm, and 7 cm. We can use the Pythagorean theorem to find the unknown angle measure. The theorem states that the sum of the squares of the two shortest sides of a right triangle is equal to the square of the longest side. In this case, the longest side is 7 cm and the two shorter sides are 5 cm and 6 cm. So, we would plug those numbers into the equation:
5 cm^2 + 6 cm^2 = 7 cm^2
The equation simplifies to 25 + 36 = 49, so we can conclude that the angle measure is 49 degrees.
The same equation can be used to find the area of the triangle. By using the formula A = (1/2)base * height, we can calculate the area of the triangle. If we take 5 cm as the base and 6 cm as the height, the area of the triangle is 15 cm^2.
In addition to finding the unknown angle measure and the area, we can also use the Pythagorean theorem to calculate the hypotenuse of the triangle. Since we already know the two shorter sides, we can solve for the longest side. We would plug in the two shorter sides into the equation and solve for the hypotenuse:
5 cm^2 + 6 cm^2 = hypotenuse^2
This equation simplifies to 25 + 36 = hypotenuse^2, so the hypotenuse is equal to 7 cm.
The Pythagorean theorem is a helpful tool for solving problems involving acute angle triangles. It can be used to find the unknown angle measure, area, and hypotenuse of the triangle.
