Question

in $\triangle abc$, $o$ is the centroid of the triangle and $ao$ is $7.8$ cm. find the length of $ay$.\
\
$\bigcirc$ $ay = 11.7$ cm\
$\bigcirc$ $ay = 3.9$ cm\
$\bigcirc$ $ay = 12$ cm\
$\bigcirc$ $ay = 1.7$ cm

Explanation:

Step1: Recall centroid property

The centroid of a triangle divides each median in a ratio of \(2:1\), where the distance from the vertex to the centroid is twice the distance from the centroid to the midpoint of the side. So, \(AO:OY = 2:1\), and \(AY=AO + OY\). Let \(OY=x\), then \(AO = 2x\). Given \(AO = 7.8\) cm, so \(2x=7.8\), which means \(x = \frac{7.8}{2}=3.9\) cm.

Step2: Calculate AY

Now, \(AY=AO + OY=7.8+3.9 = 11.7\) cm.

Answer:

AY = 11.7 cm (corresponding to the option "AY = 11.7 cm")