Question

In \(\triangle \mathrm{ABC}\), if mid points of \(\underline{\underline{A B}}\) and \(\underline{\mathrm{A} \mathrm{C}}\) are \(\mathrm{D}\) and \(\mathrm{E}\) respectively, then \(\underline{\mathrm{BE}}+\underline{\mathrm{DC}}=\) (A) \((3 / 2) \underline{\mathrm{BC}}\) (B) \((2 / 3) \underline{\mathrm{BE}}\) (C) \((3 / 2) \mathrm{BE}\) (D) \((2 / 3) \mathrm{BC}\)

Short answer

(C) \((3 / 2) \mathrm{BE}\)

Step-by-step solution

Use the midpoint theorem to find DE

Since D and E are midpoints of sides AB and AC, respectively, we know from the midpoint theorem that DE is parallel to BC, and DE = (1/2) BC.

Find the length of DC

Since D is the midpoint of side AB, then AD = (1/2) AB and BD = (1/2) AB. Similarly, since E is the midpoint of side AC, then AE = (1/2) AC and EC = (1/2) AC. Using the triangle inequality theorem, we can write DC = BD + BE + EC.

Express DC in terms of BC

We can now rewrite the expression for DC as DC = (1/2) AB + BE + (1/2) AC. Since DE = (1/2) BC, we can replace (1/2) AB + (1/2) AC with DE, giving us DC = DE + BE.

Find the sum of BE and DC

We are looking for the expression BE + DC. Since we know DC = DE + BE, we can replace DC in our expression: BE + DC = BE + (DE + BE). This simplifies to BE + DC = 2BE + DE.

Match the result with the given options

We know DE = (1/2) BC from step 1, so we can replace DE in our expression: BE + DC = 2BE + (1/2) BC. Comparing this with the given options, we see that it matches with option (C) \((3/2) BE\). So, the correct answer is: (C) \((3 / 2) \mathrm{BE}\)

Key concepts

Triangle Inequality Theorem

Understanding the Triangle Inequality Theorem is foundational when solving problems related to the lengths of sides of a triangle. Simply put, this theorem states that the sum of the lengths of any two sides of a triangle must always be greater than the length of the third side. Let's consider a triangle with sides length a, b, and c. According to this theorem, we must have:

• \( a + b > c \)
• \( a + c > b \)
• \( b + c > a \)

This concept ensures the feasibility of a triangle's existence and helps in problems where we need to find an unknown side given the lengths of the other sides. For instance, if you know the length of two sides of a triangle, you can determine a range of possible lengths for the third side. Furthermore, it assists in deriving the length of segments within triangles, as illustrated in the original exercise, where the length of segment DC is deduced by using known lengths and properties of midpoints.

Parallel Lines in Triangles

When delving into the properties of triangles, understanding how parallel lines affect their geometry is crucial. A key concept is that if a line segment joins the midpoints of two sides of a triangle, not only is this segment half the length of the third side, it is also parallel to it. Using this idea helps us connect segments within a triangle to its external sides.

For example, in the exercise given, segment DE is drawn to connect midpoints D and E, which are on sides AB and AC respectively. By invoking this concept, we know immediately that DE is parallel to side BC and is exactly half its length. This knowledge is instrumental to finding related lengths within the triangle and solving the original problem. The parallel line relationships within triangles often simplify complex geometry problems and provide a shortcut to understanding the interconnected nature of the triangle's sides.

Properties of Triangle Midsegments

A midsegment in a triangle is a segment that connects the midpoints of two sides of that triangle, and it has unique and useful properties. The main properties seen in midsegments can be summarized as follows:

• The midsegment is parallel to the third side of the triangle.
• The length of the midsegment is half the length of the third side.

These properties are a direct consequence of the midsegment theorem, which is an essential tool in solving numerous geometric problems. Applying this theorem to the exercise, we established a relationship between the midsegment DE and the side BC, which results in stating DE as (1/2) BC. This property helps us link segments within the triangle to the larger context of the triangle as a whole, providing a simple method to track and calculate distances and to understand how the triangle's parts relate to each other.