Question

In questions \(1-10\) solve \(\triangle \mathrm{ABC}\) given \(\mathrm{AB}=36 \mathrm{~cm}, \mathrm{BC}=36 \mathrm{~cm}, B=60^{\circ}\)

Short answer

Question: Given an isosceles triangle with sides AB = 36 cm, BC = 36 cm, and angle B = 60°, find the remaining angles and side lengths. Answer: In this isosceles triangle, all sides are equal at 36 cm, and all angles are equal at 60°.

Step-by-step solution

Identify given information

Triangle \(\triangle ABC\) is given with sides \(\mathrm{AB}=36\,\mathrm{cm}\), \(\mathrm{BC}=36\,\mathrm{cm}\) and angle \(B=60^\circ\). This is an isosceles triangle where \(\mathrm{AB}=\mathrm{BC}\).

Find the third side using the Law of Cosines

Apply the Law of Cosines to find side \(\mathrm{AC}\). Since we have angle \(B\) and sides \(\mathrm{AB}\) and \(\mathrm{BC}\): \[AC^2 = AB^2 + BC^2 - 2 \cdot AB \cdot BC \cdot \cos{B} \] Substitute the given values: \[AC^2 = 36^2 + 36^2 - 2 \cdot 36 \cdot 36 \cdot \cos{60^\circ} \] \[AC^2 = 1296 + 1296 - 2592 \cdot \frac{1}{2} \] \[AC^2 = 1296 \] \[AC = \sqrt{1296} = 36\, \mathrm{cm} \] So side \(\mathrm{AC} = 36\, \mathrm{cm}\).

Find the remaining angles

Since \(\mathrm{AB}=\mathrm{BC}=\mathrm{AC}\), the triangle is equilateral. Thus, all angles are equal: \[A=C=B=60^\circ\] So angles \(A\) and \(C\) are both \(60^\circ\).

Write the complete solution

We found that the triangle \(\triangle ABC\) is equilateral with all sides equal to \(36\, \mathrm{cm}\) and all angles equal to \(60^\circ\). So, the final solution is: \(\mathrm{AB}=\mathrm{BC}=\mathrm{AC}=36\, \mathrm{cm}\) \(A=B=C=60^\circ\)

Key concepts

Isosceles Triangle

An isosceles triangle is a triangle with two sides of equal length. In the context of the exercise, triangle \(\triangle ABC\) is isosceles since \(\mathrm{AB} = \mathrm{BC} = 36\, \mathrm{cm}\).
This property gives us some useful mathematical insights.

• Equal sides lead to equal opposite angles
• It simplifies calculations when solving the triangle

In our exercise, the angle \(B\) is given as \(60^{\circ}\). When an isosceles triangle has one angle known, the Law of Cosines can be used to find the third side, further aiding us in completely solving the triangle.
Understanding the nature of an isosceles triangle is crucial as it serves as a foundation to determining whether further properties, like equilateral status, are present after applying mathematical principles.

Law of Cosines

The Law of Cosines is a powerful tool in trigonometry to find unknown sides or angles in any triangle. It is particularly useful when dealing with non-right angled triangles. In our problem, the Law of Cosines helps find side \(\mathrm{AC}\) of triangle \(\triangle ABC\).
The formula is:

• \(c^2 = a^2 + b^2 - 2ab\cos(C)\)

Here, \(a\) and \(b\) are the known sides (\(\mathrm{AB}\) and \(\mathrm{BC}\)), while \(C\) is the known angle (\(B\)). Thus, substituting the values provided:

• \[ AC^2 = 36^2 + 36^2 - 2 \cdot 36 \cdot 36 \cdot \cos{60^{\circ}} \]
• \[ AC^2 = 1296 + 1296 - 2592 \cdot \frac{1}{2} \]
• \[ AC = \sqrt{1296} = 36\, \mathrm{cm} \]

This shows \(\triangle ABC\) is more than just isosceles; it's equilateral. This insight simplifies further calculations and solutions as all sides and angles are known.

Equilateral Triangle

An equilateral triangle is a triangle where all sides and all angles are equal.
In the context of the exercise, after applying the Law of Cosines, we discovered that \(\triangle ABC\) has all sides equal to \(36\, \mathrm{cm}\) and all angles equal to \(60^{\circ}\).
This makes \(\triangle ABC\) equilateral.

• Each angle in an equilateral triangle is always \(60^{\circ}\)
• All sides are of equal length, simplifying the geometric properties and calculations

Equilateral triangles often simplify geometry problems because knowing one side or angle easily leads to finding other properties. They serve as perfect examples to understand symmetry in triangles and apply fundamental trigonometric principles.