Problem 32 - Basic Technical Mathematics with Calculus

Question

Solve the given problems. If the midpoints of the sides of an isosceles triangle are joined, another triangle is formed. What do you conclude about this inner triangle?

Step-by-Step Solution

Answer

The inner triangle is equilateral.

Step 1: Understand the Problem

We are given an isosceles triangle, and we must determine the properties of the triangle formed by joining the midpoints of its sides.

Step 2: Consider the Midpoint Theorem

The Midpoint Theorem states that the segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.

Step 3: Apply the Midpoint Theorem

Consider triangle ABC with AB = AC (isosceles). Let D, E, and F be the midpoints of BC, CA, and AB respectively. By the Midpoint Theorem, DE is parallel to AB and DE = \( \frac{1}{2} \cdot AB \), EF is parallel to BC and EF = \( \frac{1}{2} \cdot BC \), and DF is parallel to AC and DF = \( \frac{1}{2} \cdot AC \).

Step 4: Conclude the Property of the Inner Triangle

Since DE, EF, and DF are parallel to AB, BC, and AC and proportional to them (half their lengths), triangle DEF is similar to triangle ABC and is equilateral as each side is equal due to the properties of an isosceles triangle being applied identically. This makes the inner triangle equilateral.

Key Concepts

Isosceles Triangle

An isosceles triangle is a special type of triangle with two sides that are of equal length, known as the 'legs'. The angle opposite the base of the triangle is called the vertex angle, and the other two angles are called the base angles. Since the two legs are equal, the base angles are also equal.

Isosceles triangles have a couple of interesting properties:

• They have reflective symmetry across the line that bisects the vertex angle, down to the midpoint of the base.
• This line also acts as the altitude, median, and angle bisector of the triangle.

Due to these symmetric properties, if we connect the midpoints of the sides of an isosceles triangle using the Midpoint Theorem, the resulting inner triangle tends to retain certain symmetry and balance, leading to further fascinating geometrical properties.

Triangle Similarity

Triangle similarity is a foundational concept in geometry where two triangles are similar if their corresponding angles are equal and their corresponding sides are in proportion.

This means that similar triangles have the same shape but may differ in size. There are specific criteria to prove triangle similarity:

• AA (Angle-Angle): If two angles of one triangle are equal to two angles of another triangle, the triangles are similar.
• SSS (Side-Side-Side): If the corresponding sides of two triangles are in proportion, the triangles are similar.
• SAS (Side-Angle-Side): If two sides of one triangle are proportional to two sides of another triangle and the included angles are equal, the triangles are similar.

When connecting the midpoints of an isosceles triangle (like in step 3 of the solution), the new smaller triangle inside is similar to the original isosceles triangle due to the proportional sides and parallel lines, thanks to the Midpoint Theorem.

Equilateral Triangle

An equilateral triangle is a triangle in which all three sides are equal in length, and consequently, all three angles are equal, each measuring 60 degrees. This makes equilateral triangles perfectly symmetrical.

The properties of equilateral triangles include:

• They are also equiangular, meaning each angle is the same.
• Any altitude not only acts as a perpendicular bisector of the opposite side but also the median and angle bisector.
• They are always similar to each other.

In the context of the problem, when the midpoints of an isosceles triangle are connected using the Midpoint Theorem, the resulting triangle is equilateral. This happens because each side of this inner triangle is formed to be half the length of the equal sides of the original triangle, maintaining side equality and forming equal angles, hence resulting in an equilateral triangle.