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Acute Angle Triangle
An acute triangle is a type of triangle where all three angles are acute angles. An acute angle is an angle that measures less than 90 degrees. In an acute triangle, each of its three angles measures less than 90 degrees.
Characteristics of an acute triangle:
All three angles are acute angles, meaning each angle is less than 90 degrees.
Since the sum of the angles in any triangle is always 180 degrees, the sum of the angles in an acute triangle is less than 180 degrees (sum of three angles < 180 degrees).
The sides opposite the acute angles are the longest sides of the triangle.
Example:
Let’s say we have an acute triangle with angles A, B, and C.
- Angle A measures 40 degrees.
- Angle B measures 60 degrees.
- Angle C measures 80 degrees.
In this example, all three angles (A, B, and C) are acute angles because they are all less than 90 degrees.
It’s worth noting that there are three other types of triangles based on their angle measurements:
What is an Acute Triangle?
An acute triangle is a type of triangle characterized by its angles. In Euclidean geometry, a triangle is a polygon with three sides and three angles. An acute triangle is one in which all three angles are acute angles, meaning each angle is less than 90 degrees.
In other words, in an acute triangle, each angle measures less than a right angle (90 degrees). Since the sum of the angles in any triangle is always 180 degrees, the sum of the three acute angles in an acute triangle is less than 180 degrees.
Here’s a quick summary of triangle types based on their angles:
- Acute Triangle: All three angles are less than 90 degrees.
- Right Triangle: One angle is exactly 90 degrees.
- Obtuse Triangle: One angle is greater than 90 degrees.
- Equiangular Triangle: All three angles are equal, and each measures 60 degrees (since 3 * 60 = 180 degrees).
Remember that the classification of a triangle based on its angles is independent of its side lengths, which are used to categorise triangles as equilateral, isosceles, or scalene.
Types of Acute Triangle
In geometry, a triangle is a three-sided polygon. An acute triangle is a type of triangle that has all three interior angles measuring less than 90 degrees. In other words, each angle in an acute triangle is acute (less than 90 degrees). Here are the types of acute triangles based on their side lengths:
- Acute Scalene Triangle: An acute triangle with all three sides of different lengths.
- Acute Isosceles Triangle: An acute triangle with two sides of equal length. The two angles opposite the equal sides are also congruent.
- Acute Equilateral Triangle: An acute triangle with all three sides of equal length. Since all angles in an equilateral triangle are equal, each angle will measure 60 degrees in an acute equilateral triangle.
It’s important to note that in an acute triangle, the sum of its three interior angles will always be less than 180 degrees (since each angle is less than 90 degrees). The sum of angles in any triangle is always 180 degrees.
Properties of Acute Triangle
An acute-angled triangle, also known as an acute triangle, is a type of triangle in which all three angles are acute angles, meaning they are less than 90 degrees. Here are some key properties of an acute-angled triangle:
Angles: All three angles of an acute triangle are less than 90 degrees. Let’s call these angles A, B, and C. So, 0° < A, B, C < 90°.
Sum of angles: The sum of the three angles in any triangle is always 180 degrees. In an acute triangle, the sum of the three acute angles is 180 degrees.
Side lengths: An acute triangle has three sides of different lengths. The lengths of the sides are usually denoted by lowercase letters a, b, and c, with their corresponding opposite angles A, B, and C.
Interior: All three vertices (corners) of an acute triangle lie within the interior of the triangle, meaning they are not on the edges or extensions of the sides.
Perimeter: The perimeter of an acute triangle is the sum of the lengths of its three sides, i.e., Perimeter = a + b + c.
Area: The area of an acute triangle can be calculated using the formula: Area = (1/2) * base * height, where the base and height are chosen relative to one of the sides, and the corresponding height forms a right angle with that side.
Altitudes: An acute triangle has three altitudes, each drawn from a vertex to the opposite side in a way that forms a right angle. The altitudes are always within the triangle.
Circumcenter: The circumcenter of an acute triangle is the center of the circle passing through all three vertices of the triangle. In an acute triangle, the circumcenter is always inside the triangle.
Incenter: The incenter of an acute triangle is the point where all three angle bisectors intersect. The angle bisectors are the lines that divide each angle into two equal angles. In an acute triangle, the incenter is always inside the triangle.
Orthocenter: The orthocenter of an acute triangle is the point where all three altitudes intersect. In an acute triangle, the orthocenter is always inside the triangle.
Remember that an acute triangle is just one of several classifications of triangles based on their angles and side lengths. Other types include right-angled triangles (one angle is 90 degrees), obtuse-angled triangles (one angle is greater than 90 degrees), and equilateral triangles (all three sides and angles are equal).
Acute Angled Triangle Formulas
An acute-angled triangle is a type of triangle where all three angles are acute, meaning each angle measures less than 90 degrees. In such triangles, the sides have specific relationships, and there are several important formulas that relate to acute-angled triangles:
Pythagorean Theorem:
In an acute-angled triangle, the Pythagorean theorem holds true:
a² + b² = c²
Here, ‘a’ and ‘b’ are the lengths of the two shorter sides (legs) of the triangle, and ‘c’ is the length of the longest side (hypotenuse).
Law of Sines:
The Law of Sines relates the side lengths of an acute-angled triangle to the sines of the angles:
a/sin(A) = b/sin(B) = c/sin(C)
Here, ‘a’, ‘b’, and ‘c’ are the side lengths, and ‘A’, ‘B’, and ‘C’ are the angles opposite their respective sides.
Law of Cosines:
The Law of Cosines relates the side lengths and angles of an acute-angled triangle:
c² = a² + b² – 2ab * cos(C)
a² = b² + c² – 2bc * cos(A)
b² = a² + c² – 2ac * cos(B)
Here, ‘a’, ‘b’, and ‘c’ are the side lengths, and ‘A’, ‘B’, and ‘C’ are the angles opposite their respective sides.
Area Formula:
The area (A) of an acute-angled triangle can be calculated using any of the following formulas:
A = 0.5 * a * b * sin(C)
A = 0.5 * b * c * sin(A)
A = 0.5 * a * c * sin(B)
Here, ‘a’, ‘b’, and ‘c’ are the side lengths, and ‘A’ is the area of the triangle.
These formulas specifically apply to acute-angled triangles, and they can be useful for solving various problems related to such triangles in geometry and trigonometry.
Perimeter of a Acute Triangle
To find the perimeter of a triangle, you need to add the lengths of all three sides. An acute triangle is a triangle in which all three angles are acute, meaning they are less than 90 degrees.
Let’s assume the sides of the acute triangle are represented by the variables ‘a’, ‘b’, and ‘c’.
Perimeter = a + b + c
Since the triangle is acute, all sides are positive lengths.
Without specific values for the sides of the triangle, we cannot calculate the actual perimeter. If you provide the lengths of the sides, I can help you find the perimeter.
Solved Problems on Acute Triangle
An acute triangle is a type of triangle in which all three angles are less than 90 degrees.
Solved Example 1:
Consider a triangle with angles A, B, and C measuring 60°, 70°, and 50°, respectively. Determine whether the triangle is an acute triangle.
Solution:
To determine if the triangle is an acute triangle, we need to check if all three angles are less than 90 degrees.
Angle A = 60° < 90° (True)
Angle B = 70° < 90° (True)
Angle C = 50° < 90° (True)
Since all three angles are less than 90 degrees, the triangle is an acute triangle.
Solved Example 2:
In an acute triangle XYZ, the lengths of the sides are as follows: XY = 5 cm, YZ = 7 cm, and XZ = 8 cm. Determine the smallest angle in the triangle.
Solution:
To find the smallest angle in the triangle, we can use the law of cosines to calculate the cosines of all three angles and then compare them.
Let’s calculate the cosines of angles:
cos(X) = (YZ^2 + XZ^2 – XY^2) / (2 * YZ * XZ)
cos(X) = (7^2 + 8^2 – 5^2) / (2 * 7 * 8)
cos(X) = (49 + 64 – 25) / 112
cos(X) = 88 / 112
cos(X) = 0.7857
cos(Y) = (XZ^2 + XY^2 – YZ^2) / (2 * XZ * XY)
cos(Y) = (8^2 + 5^2 – 7^2) / (2 * 8 * 5)
cos(Y) = (64 + 25 – 49) / 80
cos(Y) = 40 / 80
cos(Y) = 0.5
cos(Z) = (XY^2 + YZ^2 – XZ^2) / (2 * XY * YZ)
cos(Z) = (5^2 + 7^2 – 8^2) / (2 * 5 * 7)
cos(Z) = (25 + 49 – 64) / 70
cos(Z) = 10 / 70
cos(Z) = 0.1429
Now, since this is an acute triangle, the smallest angle will have the largest cosine value. In this case, cos(X) has the largest value.
The smallest angle is angle X. To find the actual angle, we can use the inverse cosine function:
X = cos^(-1)(0.7857) ≈ 38.37°
So, the smallest angle in the triangle XYZ is approximately 38.37 degrees.
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