solve the questions ;
in the given figure A, B,and C are points on OP , OQ, and OR respectively such that AB ||PQ and AC || PR show that BC|| QR .
Answers
in ∆OPQ ,
AB || PQ [ given]
OA / AP = OB / BQ (eq 1)
[ by basic proportionality theorem]
Also in ∆ OPR
AC || PR. [ GIVEN]
SO,
OA / AP = OC / CR ( eq 2)
from eq 1 and 2
OB / BQ = OC / CR
= > BC || QR
[ by converse of basic proportionality theorem]
be brainly
CONCEPT :
➦ We solve this geometry, use a similar triangle concept. Similar triangles, two figures having the same shape (but not necessarily the same size) are called similar figures.
★ The first part of the solution is important, for that, we use one of the theorems of a similar triangle that is, if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
The second part of the solution is done by using “if a line divides any two sides of a triangle in the same ratio, the line is parallel to the third side”.
REQUIRED ANSWER :
Given:
➜ AB ∥ PQ
➜ and AC ∥ PR
To show that:
★ BC ∥ QR
★ In a Δ OPQ
AB ∥ PQ
★ Here, by the theorem, in a ΔOPQ the line AB
★ which is parallel to PQ intersects the other two sides in distinct points, so it divides the other two side in same ratio
➜ The ratio is, OA / AP , OB / BQ .... ( 1 )
➜ Similarly, OC ΔOPR
➜ AC ∥ PR
➜Here, by the theorem, in a ΔOPR the line AC which is parallel to PR intersects the other two sides in distinct points, so it divides the other two side in same ratio
The ratio is, OC / CR , OA / AP ... ( 2 )
★ The ratio of (1)&(2) shows the two ΔOPQ and ΔOPR, BC ∥ QR also corresponding sides are same ratio OA / AP , OB / BQ
★ OC / CR , = OA / AP
Thus,
★ OB / BQ , OC / CR
Hence proved.
➜ Now, we can say that the triangle is a similar triangle. To show that the side is parallel, we use the second part of the hint In a ΔOQR ,
★ OC / CR , OB / BQ
➜ That is, line BC divides the triangle ΔOQR in the same ratio
Therefore, BC ∥ QR
★ Hence proved.
As We know that, The theorem, we state the first part and second part is called basic proportionality theorem.Two triangles are said to be similar, if their corresponding angles are equal and their corresponding sides are in the same ratio proportion.
