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Geometry - Classification of Triangles

Grade 5ICSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A triangle is a closed geometric shape formed by three line segments joining three non-collinear points. It has three vertices, three sides, and three interior angles. Visually, it is the simplest polygon, looking like a flat shape with three sharp corners.

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Triangles can be classified by their sides into three types: Equilateral, Isosceles, and Scalene. An Equilateral triangle has all three sides equal and looks perfectly balanced from every direction. An Isosceles triangle has two equal sides, often resembling a tall, thin peak or a wide roof. A Scalene triangle has no equal sides and appears irregular or tilted.

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Triangles can also be classified by their angles: Acute-angled, Right-angled, and Obtuse-angled. In an Acute-angled triangle, every angle is less than 90∘90^{\circ}, making every corner look 'sharp'. In a Right-angled triangle, one angle is exactly 90∘90^{\circ}, forming a perfect 'L' shape at one corner. In an Obtuse-angled triangle, one angle is greater than 90∘90^{\circ}, looking like one corner has been spread out wide.

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The Angle Sum Property is a fundamental rule stating that the sum of the three interior angles of any triangle is always 180∘180^{\circ}. If you were to cut off the three corners of any triangle and place them side-by-side, they would perfectly form a straight line.

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Properties of Equilateral Triangles: In addition to having three equal sides, every interior angle in an equilateral triangle is exactly 60∘60^{\circ}. Because 60∘60^{\circ} is less than 90∘90^{\circ}, every equilateral triangle is also an acute-angled triangle.

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Properties of Isosceles Triangles: In an isosceles triangle, the angles opposite the equal sides are also equal. For example, if two sides of a triangle are 5 cm5\text{ cm} each, the angles at the base of these sides will have the same measure.

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The Perimeter of a triangle is the total length of its boundary. Visually, if you were to place a string along the three edges of the triangle and then straighten the string, its total length would be the perimeter.

📐Formulae

Sum of interior angles: ∠A+∠B+∠C=180∘\angle A + \angle B + \angle C = 180^{\circ}

Perimeter of a triangle: P=a+b+cP = a + b + c, where a,b,ca, b, c are the lengths of the sides.

Perimeter of an Equilateral triangle: P=3×sP = 3 \times s, where ss is the length of one side.

Condition for a triangle: The sum of the lengths of any two sides must be greater than the length of the third side: a+b>ca + b > c

💡Examples

Problem 1:

In ΔPQR\Delta PQR, ∠P=55∘\angle P = 55^{\circ} and ∠Q=65∘\angle Q = 65^{\circ}. Find the measure of ∠R\angle R and classify the triangle based on its angles.

Solution:

Step 1: Use the Angle Sum Property. ∠P+∠Q+∠R=180∘\angle P + \angle Q + \angle R = 180^{\circ}.\nStep 2: Substitute the known values: 55∘+65∘+∠R=180∘55^{\circ} + 65^{\circ} + \angle R = 180^{\circ}.\nStep 3: Calculate the sum of the known angles: 120∘+∠R=180∘120^{\circ} + \angle R = 180^{\circ}.\nStep 4: Solve for ∠R\angle R: ∠R=180∘−120∘=60∘\angle R = 180^{\circ} - 120^{\circ} = 60^{\circ}.\nStep 5: Since all three angles (55∘,65∘,60∘55^{\circ}, 65^{\circ}, 60^{\circ}) are less than 90∘90^{\circ}, the triangle is an Acute-angled triangle.

Explanation:

To find a missing angle, subtract the sum of the known angles from 180∘180^{\circ}. Once all angles are known, compare them to 90∘90^{\circ} to classify the triangle.

Problem 2:

A triangle has side lengths of 8 cm8\text{ cm}, 11 cm11\text{ cm}, and 8 cm8\text{ cm}. Classify this triangle based on its sides and calculate its perimeter.

Solution:

Step 1: Identify the side lengths: s1=8 cms_1 = 8\text{ cm}, s2=11 cms_2 = 11\text{ cm}, s3=8 cms_3 = 8\text{ cm}.\nStep 2: Observe that two sides are equal (8 cm=8 cm8\text{ cm} = 8\text{ cm}). Therefore, the triangle is an Isosceles triangle.\nStep 3: Calculate the perimeter using the formula P=s1+s2+s3P = s_1 + s_2 + s_3.\nStep 4: P=8+11+8=27 cmP = 8 + 11 + 8 = 27\text{ cm}.

Explanation:

Classification by sides depends on how many sides have equal lengths. Since exactly two sides are the same, it is isosceles. The perimeter is simply the sum of all sides.