Finding the Length of PL in an Equilateral Triangle Centroid Problem

The question asks us to find the length of the segment PL, where PQR is an equilateral triangle with side length 6 cm and L is its centroid.

In an equilateral triangle, the centroid is the point where the three medians intersect. A median connects a vertex to the midpoint of the opposite side. In an equilateral triangle, the medians are also the altitudes and the angle bisectors.

Let's consider the median from vertex P to the midpoint of the side QR. Let's call this midpoint M. So, PM is a median of triangle PQR.

Since PM is also an altitude in an equilateral triangle, triangle PMR (or PMQ) is a right-angled triangle with the right angle at M. The length of QM (or MR) is half the length of QR. Since PQ = QR = RP = 6 cm, QM = \( \frac{1}{2} \times 6 \) cm = 3 cm.

We can find the length of the median PM using the Pythagorean theorem in triangle PMR:

\( PR^2 = PM^2 + MR^2 \)

\( 6^2 = PM^2 + 3^2 \)

\( 36 = PM^2 + 9 \)

\( PM^2 = 36 - 9 \)

\( PM^2 = 27 \)

\( PM = \sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3} \) cm.

Alternatively, we can use the formula for the height (which is the median length) of an equilateral triangle with side length \( s \): \( h = \frac{\sqrt{3}}{2}s \).

Here, \( s = 6 \) cm, so \( PM = \frac{\sqrt{3}}{2} \times 6 = 3\sqrt{3} \) cm.

The centroid L divides each median in the ratio 2:1, with the vertex part being twice the length of the base part. This means the distance from the vertex (P) to the centroid (L) is 2/3 of the length of the entire median (PM), and the distance from the centroid (L) to the midpoint of the side (M) is 1/3 of the median length.

So, the length of PL is \( \frac{2}{3} \) of the length of PM.

\( PL = \frac{2}{3} \times PM \)

\( PL = \frac{2}{3} \times 3\sqrt{3} \)

\( PL = 2\sqrt{3} \) cm.

Therefore, the length of PL is \( 2\sqrt{3} \) cm.

Summary of Calculation for Centroid Distance

Revision Table: Equilateral Triangle Properties & Centroid

Additional Information: Centroid and other Triangle Centers

Besides the centroid, there are other important centers in a triangle:

• Incenter: The intersection point of the angle bisectors. It is the center of the inscribed circle (incircle) of the triangle.
• Circumcenter: The intersection point of the perpendicular bisectors of the sides. It is the center of the circumscribed circle (circumcircle) that passes through all three vertices.
• Orthocenter: The intersection point of the altitudes.

In an equilateral triangle, all four of these centers (centroid, incenter, circumcenter, and orthocenter) coincide at the same point. This is a unique property of equilateral triangles (and also true for any triangle that is both equilateral and equiangular, which an equilateral triangle is).

The distance from the centroid (which is also the circumcenter) to a vertex (like PL) is the circumradius (R) of the equilateral triangle. \( R = \frac{2}{3} \times \text{median} \). The distance from the centroid (which is also the incenter) to the midpoint of a side (like LM) is the inradius (r). \( r = \frac{1}{3} \times \text{median} \). The relationship is \( R = 2r \).

Using the median length \( 3\sqrt{3} \) cm:

• Circumradius \( R = PL = \frac{2}{3} \times 3\sqrt{3} = 2\sqrt{3} \) cm.
• Inradius \( r = LM = \frac{1}{3} \times 3\sqrt{3} = \sqrt{3} \) cm.

This confirms our calculated length for PL.