Question

In the following figure \[X\] is the midpoint of the side \[QR\] and \[XZ||PQ\]and \[YZ||PX\] prove that \[YR=\dfrac{1}{4}QR\]

Answer 1

Hint: In a triangle line joining the mid point of two sides is parallel to the third side OR a line passing through midpoint of a side and parallel to the other side goes through the mid point of the third side.

Complete step-by-step answer:

As given \[X\] is mid point of \[QR\] and \[XZ||PQ\] then \[Z\] is the midpoint of \[PR\].
Now again, as \[Z\] is the midpoint of \[PR\] and \[YZ||PX\]it can be concluded that \[Y\] is the midpoint of \[XR\].
Now we have \[X\] is mid point of \[QR\] hence \[XR=\dfrac{1}{2}QR\]
Now as \[Y\] is the midpoint of \[XR\] we have \[YR=\dfrac{1}{2}XR\].
$\Rightarrow$ Now using the above two relation we have \[YR=\dfrac{1}{2}\times \dfrac{1}{2}QR\]\[\Rightarrow YR=\dfrac{1}{4}QR\]

Note: Two theorems can be used to prove the above problem.
1.In a triangle line joining the mid point of two sides is parallel to the third side.
2. In a triangle if a line parallel to one side of the triangle and passing through the midpoint of another side will bisect the third side.