If in ΔABC, AB = 5 cm, BC = 12 cm, and AC = 13 cm, then the length of the median BE is:

Question

Prepp Team · Accepted Answer

Finding the Length of a Median in a Triangle

We are given a triangle ABC with the following side lengths:

AB = 5 cm
BC = 12 cm
AC = 13 cm

We need to find the length of the median BE, where E is the midpoint of side AC.

Identifying the Type of Triangle

First, let's check if the triangle ABC is a right-angled triangle. We can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

Let's calculate the squares of the side lengths:

\(AB^2 = (5 \text{ cm})^2 = 25 \text{ cm}^2\)
\(BC^2 = (12 \text{ cm})^2 = 144 \text{ cm}^2\)
\(AC^2 = (13 \text{ cm})^2 = 169 \text{ cm}^2\)

Now, let's check if the sum of the squares of the two shorter sides equals the square of the longest side:

\(AB^2 + BC^2 = 25 \text{ cm}^2 + 144 \text{ cm}^2 = 169 \text{ cm}^2\)

We see that \(AB^2 + BC^2 = AC^2\) because \(169 \text{ cm}^2 = 169 \text{ cm}^2\). This confirms that triangle ABC is a right-angled triangle, with the right angle at vertex B (the vertex opposite the hypotenuse AC).

Length of the Median to the Hypotenuse

In a right-angled triangle, the median drawn from the vertex with the right angle to the hypotenuse has a special property. The length of this median is exactly half the length of the hypotenuse.

In our triangle ABC, BE is the median from the right angle vertex B to the hypotenuse AC. Therefore, the length of the median BE is half the length of AC.

\(\text{Length of median BE} = \frac{1}{2} \times \text{Length of hypotenuse AC}\)

\(\text{BE} = \frac{1}{2} \times 13 \text{ cm}\)

\(\text{BE} = 6.5 \text{ cm}\)

Summary of Calculation

Step	Description	Calculation/Result
1	Check for right triangle using Pythagorean theorem	\(5^2 + 12^2 = 25 + 144 = 169\) \(13^2 = 169\) \(5^2 + 12^2 = 13^2\), so right triangle at B
2	Identify hypotenuse	AC (opposite the right angle) = 13 cm
3	Apply property of median to hypotenuse	Median BE = \(\frac{1}{2} \times \text{Hypotenuse AC}\)
4	Calculate median length	BE = \(\frac{1}{2} \times 13 \text{ cm} = 6.5 \text{ cm}\)

The length of the median BE is 6.5 cm.

Revision Table: Triangle Medians and Properties

Concept	Definition	Property in Right Triangle
Median	A line segment joining a vertex of a triangle to the midpoint of the opposite side.	Three medians exist for any triangle.
Median to Hypotenuse (in Right Triangle)	The median drawn from the right-angle vertex to the hypotenuse.	Its length is half the length of the hypotenuse. The midpoint of the hypotenuse is the circumcenter of the right triangle.
Pythagorean Theorem	In a right triangle, \(a^2 + b^2 = c^2\), where c is the hypotenuse.	Used to check if a triangle is right-angled or to find unknown side lengths.

Additional Information: Triangle Geometry and Medians

Medians are important line segments in triangles. They have several interesting properties:

Centroid: The three medians of a triangle intersect at a single point called the centroid. The centroid is the geometric center of the triangle and is also the center of mass.
Median divides area: Each median divides the triangle into two smaller triangles of equal area.
Centroid divides median: The centroid divides each median in a 2:1 ratio, with the longer segment being between the vertex and the centroid.
Apollonius's Theorem: For any triangle ABC, if \(m_c\) is the length of the median from C to AB, then \(AC^2 + BC^2 = 2(m_c^2 + (\frac{1}{2}AB)^2)\). This theorem can be used to find the length of any median if the side lengths are known, and it provides a general method even if the triangle is not right-angled. In our case, for median BE to side AC: \(AB^2 + BC^2 = 2(BE^2 + (\frac{1}{2}AC)^2)\). Substituting the values: \(5^2 + 12^2 = 2(BE^2 + (\frac{1}{2} \times 13)^2)\) > \(25 + 144 = 2(BE^2 + 6.5^2)\) > \(169 = 2(BE^2 + 42.25)\) > \(84.5 = BE^2 + 42.25\) > \(BE^2 = 84.5 - 42.25 = 42.25\) > \(BE = \sqrt{42.25} = 6.5\) cm. This confirms the result obtained using the right-triangle property.