Scalene Triangle

Scalene Triangle Scalene Triangle As Geometry is the study of various shapes and their properties, we come across different types of figures and structures. Among them, triangles are one of the most commonly observed shapes. A triangle belongs to the polygon family of shapes and hence just like other polygons, it is also a 2-dimensional closed figure bounded by straight line sides. As the name suggests, tri-angle implies three angled. So a triangle is a figure which consists of 3 sides and hence 3 angles. We observe that triangles are classified into different types based on their sides measurement and their angles measurement. Based on triangles angles, we have 3 categories as shown below: a) Acute-angled triangle: A triangle in which all the 3 angles measure less than 90. b) Obtuse-angled triangle: A triangle which has one angle greater than 90, with other two angles less than 90. (Since the sum of all the 3 angles in any triangle must be equal to 180, hence there cannot be more than one obtuse angle in a triangle). c) Right-angled triangle: A triangle in which one of the angles is equal to 90. Based on triangles sides, we have 3 categories as shown below: a) Equilateral triangle: All the 3 sides of the triangle are equal to each other. In this triangle, all the 3 angles are also equal to each other. Each angle measure 60. b) Isosceles triangle: Any two sides of the triangle are equal to each other. Hence, the angles opposite to the equal sides are also equal. c) Scalene Triangle: A triangle in which all the 3 sides (and hence 3 angles) are not equal to each other. Definition of a Scalene Triangle:A Scalene triangle is a triangle in which all the 3 sides do not equal each other in measurement. As the sides are not congruent to each other, hence the 3 angles are also not equal to each other. In the figures below, we can see that the triangle consists of 3 unequal sides. Properties of a Scalene Triangle: 1) A scalene triangle has no two sides equal to each other. 2) A scalene triangle has all the 3 angles unequal to each other. 3) Sum of all the angles in any triangle = 180 4) A scalene triangle does not have any line of symmetry. Example: Given that two angles of a scalene triangle PQR are equal to 56 and 88 as shown in the figure on the right. What is the measure of the third angle of the triangle? Given that angle R = 56 and angle Q = 88 Sum of all the angles of a triangle = 180 Hence, angle P + angle Q + angle R = 180 So, angle P + 88 + 56 = 180 Angle P + 144 = 180 == angle P = 180 - 144 Therefore, angle P = 36 Example: If the angles of a scalene triangle are (2x+ 20), (x + 50) and (x + 10), then what is the measure of each angle of the triangle? Sum of all angles of a triangle = 180 Hence, 2x + 20 + x + 50 + x + 10 = 180 == 4x + 80 = 180 == 4x = 100 This gives: x = 100/4 == x = 25 Now the angles are 2x + 20 = 2*25 + 20 = 70 x + 50 = 25 + 50 = 75 and x+ 10 = 25 + 10 = 35 Therefore, the 3 angles are 70, 75 and 35. Triangle Inequality Theorem: The Triangle Inequality Theorem states that any side of a triangle is always shorter than the sum of the other two sides. This applies for every triangle, including the Scalene Triangle. If a, b and c are the side lengths of the three sides of a triangle, then according to the theorem: a b + c b c + a c a + b This theorem can be understood by looking at two cases: I. If one side is longer than the sum of the other two sides, then the other two sides will not meet each other to form a triangle, as shown in the figure below. II. If one side is equal to the sum of the other two sides, then a triangle is not formed. In fact, this happens when the three points are collinear (points which lie on the same line). Therefore for a triangle to be formed, the sum of any two sides of the triangle must always be greater than the third side! Example: Can a triangle be formed with the side lengths 12m, 18m and 22m? Let the side lengths be a = 12m, b = 18m and c = 22m. Now, 12 18 + 22. Hence a b + c 18 22 + 12. Hence b c + a 22 12 + 18. Hence c a + b Since the Triangle Inequality Theorem is satisfied, hence yes with the given side lengths we can form a triangle. Perimeter of a Scalene Triangle: Perimeter of any triangle is the sum of all the sides of the triangle. Therefore, to calculate the perimeter of a scalene triangle we have to simply add up all the side lengths of the given triangle. If the lengths of the sides of a scalene triangle are a, b and c: Then the Perimeter of the Scalene triangle = a + b + c Example: Calculate the perimeter of a scalene triangle whose side lengths are 12m, 16m and 17m. Perimeter of a Scalene triangle = Sum of all the side lengths of the triangle. Perimeter = 12m + 16m + 17m = 45m Therefore, perimeter of the given scalene triangle is 45m. Area of a Scalene triangle: Area of a triangle is the space occupied within the boundaries of a triangle. This area can be calculated by using the following methods: 1) Area Formula: If the base length of a triangle and the height (which is the perpendicular line drawn from the vertex to the opposite side) of a triangle are given, then we can find the area of a triangle. If the base of a triangle is b and height of the triangle is h, then: Area of a triangle = 1/2 * base * height == Area = 1/2 * b * h 2) Herons Formula: If the side lengths of a triangle are given, then the area of a triangle can be calculated using the Herons Formula. If a triangle has side lengths as a, b and c, then the semi-perimeter, s = (a + b+ c)/2 Then, Area of the triangle = [s* (s-a)* (s-b)* (s-c)] Example: Find the area of a triangle given side QR is 10m, and the perpendicular PN is 7.5m. Given the base QR = 10m Height is the perpendicular line drawn from the vertex to the opposite side of a triangle. Since PN is the perpendicular line from the vertex P to the opposite base side QR, hence height is PN = 7.5m. Area of a triangle = 1/2 * base * height = 1/2 * 10m * 7.5m = 37.5m2 Hence, Area of the given triangle is 37.5 square meters. Example: In triangle XYZ, side XY is 5m, side YZ is 8m and side XZ is 7m a shown in the figure on the right. Is XYZ a scalene triangle? What is the area of the triangle XYZ? Given the side lengths of triangle XYZ. Since the lengths of the sides are not equal to each other, hence XYZ is a scalene triangle. Let a = 5m, b= 7m and c = 8m. Then s = (a + b + c)/2 = (5 + 7 + 8)/2 = 10 Now according to Herons formula: Area of the triangle, A = [s* (s-a)* (s-b)* (s-c)] Hence, Area = [10* (10- 5)* (10- 7)* (10- 8)] = (10 * 5 * 3 * 2) = 300 = 17.3m2 (approximately) Therefore, the area of the triangle XYZ is 17.3 square meters.