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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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Triangles are classified based on the length of their sides: Scalene (all sides unequal), Isosceles (two sides equal), and Equilateral (all three sides equal).

Diagram showing Scalene, Isosceles, and Equilateral triangles side-by-side.
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Triangles are also classified by their angles: Acute (all angles <90∘< 90^{\circ}), Right (one angle =90∘= 90^{\circ}), and Obtuse (one angle >90∘> 90^{\circ}).

Visual representation of a right-angled triangle and an obtuse-angled triangle.
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In an equilateral triangle, all three interior angles are equal, and each measures 60∘60^{\circ}.

Equilateral triangle with all angles marked as 60 degrees.
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In an isosceles triangle, the angles opposite to the equal sides are also equal.

📐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.

Problem 3:

Identify the type of triangle ABCABC where side AB=6 cmAB = 6\text{ cm}, BC=6 cmBC = 6\text{ cm}, and AC=6 cmAC = 6\text{ cm}. Find its perimeter.

An equilateral triangle ABC with all sides labeled 6 cm.

Solution:

Side AB=6 cm\text{Side } AB = 6\text{ cm} Side BC=6 cm\text{Side } BC = 6\text{ cm} Side AC=6 cm\text{Side } AC = 6\text{ cm} Since all sides are equal, it is an Equilateral triangle.\text{Since all sides are equal, it is an Equilateral triangle.} Perimeter=3×s\text{Perimeter} = 3 \times s Perimeter=3×6=18 cm\text{Perimeter} = 3 \times 6 = 18\text{ cm}

Explanation:

A triangle with all three sides of equal length is called an equilateral triangle. The perimeter is the sum of all sides, which for an equilateral triangle is 33 times the side length.

Problem 4:

In triangle XYZXYZ, ∠X=90∘\angle X = 90^{\circ}, XY=3 cmXY = 3\text{ cm}, and XZ=4 cmXZ = 4\text{ cm}. Classify the triangle based on its angles and find the third side YZYZ if the perimeter is 12 cm12\text{ cm}.

A right-angled triangle XYZ with sides 3cm and 4cm marked.

Solution:

Since ∠X=90∘, the triangle is a Right-angled triangle.\text{Since } \angle X = 90^{\circ}\text{, the triangle is a Right-angled triangle.} Perimeter=XY+XZ+YZ\text{Perimeter} = XY + XZ + YZ 12=3+4+YZ12 = 3 + 4 + YZ 12=7+YZ12 = 7 + YZ YZ=12−7=5 cmYZ = 12 - 7 = 5\text{ cm}

Explanation:

The presence of a 90∘90^{\circ} angle makes it a right-angled triangle. To find the missing side, subtract the sum of the known sides from the total perimeter.