Question

In the figure given below, point P and Q are mid points on the side AC and BP respectively. Area of each part is shown in the figure, then find the value of \[x+y\].

Answer 1

Hint: In this problem, we have to find the value of \[x+y\] where in the given figure point P and Q are mid points on the side AC and BP respectively. Area of each part is shown in the figure. We can first find the area of triangle APB and CPB, where P is the midpoint and we will get an equation, we can then find the area of triangle QCB and CPQ, where Q is the midpoint and we will get another equation, we can solve both the equations to get the value of \[x+y\].

Complete step by step solution:
Here we have to find the value of, \[x+y\] where in the given figure point P and Q are midpoints on the side AC and BP respectively.
Area of each part is shown in the figure below.

We can now find the area of the triangle APB and CPB.
We know that the Area of the triangle formula is \[\dfrac{1}{2}\times l\times b\].
We can now assume the perpendicular distance of the side AC be 2h.
We can see that for triangle APB, the length, l= y+3, Breadth, b = h.
The area of triangle APB = \[\dfrac{1}{2}\times \left( y+3 \right)h\]…….(1)
We can see that for triangle CPB, the length, l= x+7, Breadth, b = h.
The area of triangle APB = \[\dfrac{1}{2}\times \left( x+7 \right)h\]……..(2)
Since. P is the midpoint of AC, we will have AP = CP.
So, both areas will be equal,
\[\begin{align}
  & \Rightarrow \dfrac{1}{2}\left( y+3 \right)h=\dfrac{1}{2}\left( x+7 \right)h \\
 & \Rightarrow x+7=y+3 \\
 & \Rightarrow x+4=y.....(3) \\
\end{align}\]
We can now assume the perpendicular distance CR as 2t, the CQ will be t, we get
Area of triangle QCB = \[\dfrac{1}{2}\times 7t\]
Area of triangle CPQ = \[\dfrac{1}{2}xt\]
Since, Q is the midpoint of BP, we have BQ = QP,
\[\Rightarrow x=7....(4)\]
 We can now substitute (4) in (3), we get
\[\Rightarrow y=11\]…… (5)
We can now add (4) and (5), we get
\[\Rightarrow x+y=7+11=18\]
Therefore, the value of \[x+y=18\].

Note: We should always remember that the area of the triangle can be found using the formula \[\dfrac{1}{2}\times l\times b\] where l and b are the length and breadth of the given triangle. We should also know that the point P is the midpoint of AC, then we will have AP = CP, in this problem.