Answer:
It can be proved by:
The two triangles are similar and the ratio of corresponding sides are equal.
Step-by-step explanation:
We are given a ΔQXY which has a line RS inside it such that RS || XY.
To prove:
[tex]\dfrac{XR}{RQ} = \dfrac{YS}{SQ}[/tex]
First of all, let us have a look at the ΔQXY and ΔQRS as per the given question figure:
1. ∠1=∠3 (Because sides XY || RS and ∠1, ∠3 are corresponding angles)
2. ∠2=∠4 (Because sides XY || RS and ∠2, ∠4 are corresponding angles)
3. ∠Q is common to both the triangles.
So, all the three angles are equal to each other, hence the two triangles are similar:
[tex]\triangle QXY \sim \triangle QRS[/tex]
If two triangles are similar, then ratio of their corresponding sides is also equal.
[tex]\dfrac{XQ}{RQ} = \dfrac{YQ}{SQ}[/tex]
Subtracting 1 from both the sides:
[tex]\dfrac{XQ}{RQ} -1= \dfrac{YQ}{SQ}-1\\\dfrac{XQ-RQ}{RQ}= \dfrac{YQ-SQ}{SQ}[/tex]
Now, it is clearly observable that:
XQ - RQ = XR and
YQ - SQ = YS
Putting the values in above equation:
[tex]\dfrac{XR}{RQ} = \dfrac{YS}{SQ}[/tex]
Hence proved.
