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A Tale of Three Intersecting Lines - Equilateral Triangles

Grade 7CBSE

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

🔑Concepts

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An equilateral triangle is formed when three lines intersect such that the internal angles are all equal to 60∘60^\circ. By the Angle Sum Property, since the total is 180∘180^\circ, each angle must be 180∘÷3=60∘180^\circ \div 3 = 60^\circ.

Equilateral triangle with all internal angles labeled as 60 degrees.
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In an equilateral triangle, all three sides are of equal length (ss). If any one side is known, the other two sides and the total perimeter can be determined immediately.

Triangle with all sides labeled with the variable s representing equal length.
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The perimeter of an equilateral triangle is the total boundary length, calculated as P=s+s+s=3sP = s + s + s = 3s. Conversely, the side length is s=P3s = \frac{P}{3}.

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Three intersecting lines can form different types of triangles; however, if the lines are positioned such that the exterior angle is 120∘120^\circ at every vertex, the interior triangle is guaranteed to be equilateral.

📐Formulae

Sum of interior angles=180∘\text{Sum of interior angles} = 180^\circ

Each angle in an equilateral triangle=180∘3=60∘\text{Each angle in an equilateral triangle} = \frac{180^\circ}{3} = 60^\circ

Perimeter of an equilateral triangle=3s\text{Perimeter of an equilateral triangle} = 3s

Side length (s)=Perimeter3\text{Side length } (s) = \frac{\text{Perimeter}}{3}

💡Examples

Problem 1:

In an equilateral triangle △ABC\triangle ABC, if the length of side ABAB is 7 cm7\text{ cm}, find the perimeter of the triangle.

Solution:

P=3×7=21 cmP = 3 \times 7 = 21\text{ cm}

Explanation:

Since all sides of an equilateral triangle are equal, AB=BC=CA=7 cmAB = BC = CA = 7\text{ cm}. The perimeter is the sum of all sides: 7+7+7=21 cm7 + 7 + 7 = 21\text{ cm}.

Problem 2:

Find the value of xx if the angles of a triangle formed by three intersecting lines are xx, xx, and 60∘60^\circ, and it is given that the triangle is equilateral.

Solution:

x=60∘x = 60^\circ

Explanation:

In an equilateral triangle, all interior angles must be equal to 60∘60^\circ. Therefore, if one angle is 60∘60^\circ and the triangle is equilateral, the other two angles xx must also be 60∘60^\circ.

Problem 3:

The perimeter of an equilateral triangle is 45 cm45\text{ cm}. Calculate the length of each side.

Solution:

45÷315\begin{array}{r} 45 \div 3 \\ \hline 15 \end{array}

Explanation:

To find the side of an equilateral triangle when the perimeter is known, we use the formula s=P3s = \frac{P}{3}. Substituting the values: s=453=15 cms = \frac{45}{3} = 15\text{ cm}.

Problem 4:

Given an equilateral triangle PQRPQR where one side PQPQ is (2x−5) cm(2x - 5)\text{ cm} and another side QRQR is 11 cm11\text{ cm}, find the value of xx.

Equilateral triangle PQR with side PQ labeled 2x-5 and QR labeled 11cm.

Solution:

In an equilateral triangle, all sides are equal. PQ=QRPQ = QR 2x−5=112x - 5 = 11 Add 55 to both sides: 2x=11+52x = 11 + 5 2x=162x = 16 Divide by 22: x=162x = \frac{16}{2} x=8x = 8

Explanation:

Since the triangle is defined as equilateral, we set the expression for side PQPQ equal to the known length of side QRQR and solve for the unknown variable xx.

Problem 5:

A wire is bent into the shape of an equilateral triangle with a side length of 12 cm12\text{ cm}. If the same wire is straightened and then bent into a square, what would be the length of each side of the square?

Diagram showing an equilateral triangle with side 12cm being transformed into a square with unknown side length.

Solution:

Step 1: Find the total length of the wire (Perimeter of the triangle). P=3×side=3×12=36 cmP = 3 \times \text{side} = 3 \times 12 = 36\text{ cm} Step 2: The perimeter of the square will be the same as the wire length. Perimeter of square=4×side of square=36 cm\text{Perimeter of square} = 4 \times \text{side of square} = 36\text{ cm} Step 3: Solve for the side of the square. Side=364=9 cm\text{Side} = \frac{36}{4} = 9\text{ cm}

Explanation:

The length of the wire remains constant. First, we find the perimeter of the equilateral triangle, then divide that total length by 4 to find the side of a square.