Understanding the Problem: Finding the Centroid Distance

The question asks us to find the length of the segment from vertex B to the centroid O in a triangle ABC, given the lengths of its sides AB, BC, and AC.

We are given:

• AB = 48 cm
• BC = 55 cm
• AC = 73 cm
• O is the centroid of ΔABC.

We need to find the length of BO, rounded to one decimal place.

What is a Centroid?

The centroid of a triangle is the point where the three medians of the triangle intersect. A median is a line segment joining a vertex to the midpoint of the opposite side.

A key property of the centroid is that it divides each median in a 2:1 ratio. The longer segment is from the vertex to the centroid, and the shorter segment is from the centroid to the midpoint of the opposite side. For the median from vertex B (let's call the midpoint of AC as D), the centroid O lies on BD such that BO : OD = 2 : 1. This means BO = $\frac{2}{3}$ BD.

Finding the Length of the Median BD

To find BO, we first need to find the length of the median BD, which goes from vertex B to the midpoint D of side AC.

We can use the median length formula or Apollonius's Theorem. The formula for the length of the median $m_b$ from vertex B to side AC is:

$\qquad m_b = \frac{1}{2} \sqrt{2(a^2 + c^2) - b^2}$

where $a$ is the length of the side opposite vertex A (BC), $c$ is the length of the side opposite vertex C (AB), and $b$ is the length of the side opposite vertex B (AC).

In our case:

• $a = BC = 55$ cm
• $c = AB = 48$ cm
• $b = AC = 73$ cm

Substituting these values into the formula:

$\qquad BD = \frac{1}{2} \sqrt{2(55^2 + 48^2) - 73^2}$

$\qquad BD = \frac{1}{2} \sqrt{2(3025 + 2304) - 5329}$

$\qquad BD = \frac{1}{2} \sqrt{2(5329) - 5329}$

$\qquad BD = \frac{1}{2} \sqrt{10658 - 5329}$

$\qquad BD = \frac{1}{2} \sqrt{5329}$

Let's calculate $\sqrt{5329}$. $70^2 = 4900$, $80^2 = 6400$. It ends in 9, so the root must end in 3 or 7. Let's try 73.

$73 \times 73 = (70+3)(70+3) = 70^2 + 2 \times 70 \times 3 + 3^2 = 4900 + 420 + 9 = 5329$.

So, $\sqrt{5329} = 73$.

$\qquad BD = \frac{1}{2} \times 73 = 36.5$ cm.

Checking for a Right Triangle Property

Alternatively, before using the median formula, we could check if the triangle is a right-angled triangle using the Pythagorean theorem. If $a^2 + c^2 = b^2$, or $c^2 + b^2 = a^2$, or $a^2 + b^2 = c^2$, then it's a right triangle.

Let's check if the sum of the squares of the two smaller sides equals the square of the longest side (73 cm).

Smallest side: 48 cm (AB)

Middle side: 55 cm (BC)

Longest side: 73 cm (AC)

Is $48^2 + 55^2 = 73^2$?

$48^2 = 2304$

$55^2 = 3025$

$2304 + 3025 = 5329$

$73^2 = 5329$

Since $48^2 + 55^2 = 73^2$, i.e., $AB^2 + BC^2 = AC^2$, the triangle ABC is a right-angled triangle with the right angle at vertex B.

In a right-angled triangle, the median drawn to the hypotenuse is half the length of the hypotenuse. The median from vertex B is drawn to the side AC, which is the hypotenuse in this right triangle. Therefore, the length of the median BD is exactly half the length of AC.

$\qquad BD = \frac{1}{2} AC = \frac{1}{2} \times 73 = 36.5$ cm.

This confirms the result obtained using the median length formula.

Calculating the Length of BO

Now that we have the length of the median BD, we can find the length of BO using the centroid property that BO = $\frac{2}{3}$ BD.

$\qquad BO = \frac{2}{3} \times 36.5$

$\qquad BO = \frac{2}{3} \times \frac{73}{2}$

$\qquad BO = \frac{73}{3}$

Now, we perform the division:

$\qquad BO \approx 24.333...$

Rounding to One Decimal Place

The question asks for the length of BO correct to one decimal place. Rounding 24.333... to one decimal place gives 24.3 cm.

Final Answer Summary

Given a triangle ABC with sides 48 cm, 55 cm, and 73 cm, we found that it is a right-angled triangle at B. The centroid O divides the median BD (from B to the midpoint of AC) in a 2:1 ratio. The median to the hypotenuse AC is half the hypotenuse length, so BD = 36.5 cm. Therefore, BO = $\frac{2}{3}$ BD = $\frac{2}{3} \times 36.5 = \frac{73}{3} \approx 24.333...$ cm. Rounded to one decimal place, the length of BO is 24.3 cm.

Revision Table: Key Concepts

Additional Information: Triangle Centroid and Medians

The centroid is one of the four main centers of a triangle (the others being the circumcenter, orthocenter, and incenter). It is always located inside the triangle.

The centroid is also the center of mass of the triangle if the mass is uniformly distributed. This means if you cut a triangle out of a uniform material, you could balance it perfectly at its centroid.

The medians of a triangle divide it into six smaller triangles of equal area.

If the coordinates of the vertices of a triangle are $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$, the coordinates of the centroid O are given by:

$\qquad O = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right)$

While we used side lengths here, coordinate geometry provides another way to work with the centroid if vertex coordinates are known.