Question

The three angles of a triangle are in AP If the largest angle is twice the smallest angle, then the largest angle is (a) \(40^{\circ}\) (b) \(60^{\circ}\) (c) \(100^{\circ}\) (d) \(80^{\circ}\)

Short answer

Answer: The largest angle is \(90^{\circ}\).

Step-by-step solution

Assign variables to angles

Let the three angles of the triangle be \(A, B,\) and \(C\). Since they are in AP, let \(A=a\), \(B=a+d\), and \(C=a+2d\).

Apply the angle sum property

The sum of a triangle's angles is always \(180^{\circ}\). So, we can write: \(a + (a+d) + (a+2d) = 180\)

Simplify the equation

Combine the terms on the left side of the equation: \(3a + 3d = 180\)

Divide the equation by 3

We can simplify the equation further by dividing both sides by 3: \(a + d = 60\)

Use the given information about the largest angle

We are given that the largest angle (\(C\)) is twice the smallest angle (\(A\)). So, we can write: \(C = 2A\) Substituting the expressions for \(A\) and \(C\): \(a + 2d = 2a\)

Solve for \(d\)

Rearrange the last equation to solve for \(d\): \(d = a\)

Substitute \(d\) into the equation for the sum of angles

We found that \(d = a\), so substitute it back into the angle sum equation: \(a + a = 60\)

Solve for \(a\)

Combine the terms on the left side of the equation and then divide both sides by 2: \(2a = 60\) \(a = 30\)

Find the largest angle

Since we now have the value for \(a\), we can find the value for the largest angle, \(C\): \(C = a + 2d = a + 2a = 3a\) Substitute the value of \(a\): \(C = 3(30) = 90\) So the largest angle is \(90^{\circ}\). However, this answer was not given as an option, so there might be a mistake in the question. Nevertheless, this is the correct solution based on the given information.

Key concepts

Arithmetic Progression

Arithmetic progression (AP) is a sequence of numbers in which the difference between the consecutive terms is constant. This difference is called the 'common difference'. In the context of geometry, particularly with triangle angles, if the angles are in AP, they follow a pattern where each angle can be expressed in terms of the smallest angle and the common difference.

For instance, if the smallest angle is denoted by 'a' and the common difference by 'd', then the other angles can be represented as 'a + d' and 'a + 2d'. It's essential to approach problems involving AP by first identifying the pattern, setting up the terms with variables, and creating equations based on the defined properties of an arithmetic progression.

In the provided problem, knowing this concept allows us to set up our angles accurately and leads us to a simplified equation we can solve.

Angle Sum Property

The angle sum property is a fundamental principle in triangle geometry, stating that the sum of the interior angles of a triangle is always equal to 180 degrees. This property is critical for solving many geometric problems because it provides a reliable equation that must always hold true in any triangle, regardless of its shape or size.

When confronted with problems involving the interior angles of a triangle, leveraging this property helps us construct a very specific equation: \(a + b + c = 180^\circ\), where 'a', 'b', and 'c' represent the angles of the triangle. This equation is the basis of our approach to finding the value of each angle when given certain conditions, such as the angles forming an arithmetic progression, as in the textbook exercise.

Understanding and applying the angle sum property is a pillar in solving triangle-related problems and is a foundational technique for any student studying geometry.

Geometric Equations Solving

Solving geometric equations involves identifying the relationships and properties of shapes to set up and manipulate algebraic expressions. The process typically combines geometric principles, such as the angle sum property of triangles, with algebraic techniques to find the values of unknown variables.

To solve geometric equations effectively, one must thoroughly understand how to manipulate equations, including simplifying expressions, isolating variables, and employing substitution when necessary. In our example, the steps to solve for the angles of the triangle in AP included these techniques to derive the value of the smallest angle and then extended the solution to find the largest angle.

A methodical approach, starting from setting up the problem with clear variable representation, progressing to applying the relevant geometric properties, and finally using algebraic steps to reach the solution, is the core of solving geometric equations. This process not only results in finding the numerical solutions but also enhances problem-solving skills that are vital in advanced mathematics.