
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
473.7k+ views
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.
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.
Recently Updated Pages
Glucose when reduced with HI and red Phosphorus gives class 11 chemistry CBSE

The highest possible oxidation states of Uranium and class 11 chemistry CBSE

Find the value of x if the mode of the following data class 11 maths CBSE

Which of the following can be used in the Friedel Crafts class 11 chemistry CBSE

A sphere of mass 40 kg is attracted by a second sphere class 11 physics CBSE

Statement I Reactivity of aluminium decreases when class 11 chemistry CBSE

Trending doubts
When people say No pun intended what does that mea class 8 english CBSE

In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE

How many ounces are in 500 mL class 8 maths CBSE

Which king started the organization of the Kumbh fair class 8 social science CBSE

What is BLO What is the full form of BLO class 8 social science CBSE

Advantages and disadvantages of science
