Question

In the following figure drawn, line $PY\parallel BC$, $AP = 6,PB = 12,AY = 5,YC = x$, then find $x$.

Answer 1

Hint: In this question, the line $PY\parallel $side$BC$. So, by using BPT (Basic Proportionality Theorem) because BPT (Basic Proportionality Theorem) states that if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct two points, then the other two sides are divided in the same ratio.
So, here in this question we can say that $\dfrac{{AY}}{{YC}} = \dfrac{{AP}}{{PB}}$
Here, all the values are given in the question. So, we just need to put the corresponding values to get the answer.

Complete step-by-step answer:
First of all we will start solving this question by enlisting the given values.
Here in this question it is given that there is a triangle $\vartriangle ABC$ in which, line $PY\parallel BC$,
The lengths of sides are $AP = 6,PB = 12,AY = 5,YC = x$.
So, now as we know that line $PY$ is parallel to side $BC$
So, we will solve this question by using BPT (Basic Proportionality Theorem)
Because BPT (Basic Proportionality Theorem) states that if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct two points, then the other two sides are divided in the same ratio.
So, we can say that $\dfrac{{AY}}{{YC}} = \dfrac{{AP}}{{PB}}$
Now, here we know that
$AP = 6,PB = 12,AY = 5,YC = x$ …………….{Given}
So, by putting corresponding values we will get
$
  \dfrac{5}{x} = \dfrac{6}{{12}} \\
   \Rightarrow \;5\left( {12} \right) = 6x \\
   \Rightarrow 6x = 60 \\
   \Rightarrow x = 10\; \\
 $

$x = 10$ is the answer to this question.

Note: The alternative method to solve this question is by using the properties of similar triangles, for that you have to firstly prove the triangle $\vartriangle ABC$ is similar to $\vartriangle APY$ then by using the properties of similar triangle we can get the value of $x$.