in the given figure XY is a line parallel to side QR of triangle PQR if AQ is parallel to PR and BR is parallel to PQ then prove that area (APQ) is equal to area of triangle (BPR)
Answers
If XY is a line parallel to side QR of triangle PQR and AQ is parallel to PR and BR is parallel to PQ then the area (APQ) is equal to area of triangle (BPR).
Step-by-step explanation:
It is given that,
XY // QR
AQ // PR
BR // PQ
Join the points A and B to P.(as shown in the figure attached below)
Step 1:
We have,
XY // QR, so we can also write AY // QR
And,
AQ // PR, so we can also write AQ // RY
∴ Quadrilateral AQRY is a parallelogram
We know that a triangle and a parallelogram are on the same base and between the same parallel line then the area of the triangle is equal to half the area of the parallelogram.
Here, ∆APQ and parallelogram AQRY are on the same base AQ and between same parallel lines AQ & PR, therefore, we get
Area of ∆APQ = ½ * [area of parallelogram AQRY] ……. (i)
Step 2:
We also have,
XY // QR, so we can also write BX // QR
And,
BR // PQ, so we can also write BR// QX
∴ Quadrilateral BRQX is a parallelogram
Here, ∆BPR and parallelogram BRQX are on the same base BR and between same parallel lines BR & QX, therefore, we get
Area of ∆BPR = ½ * [area of parallelogram BRQX] ……. (ii)
Step 3:
We know that when two parallelograms are on the same base and between the same parallel lines then the area of the two parallelograms are equal.
Therefore,
[area of parallelogram AQRY] = [area of parallelogram BRQX] …… (iii)
Thus, from (i), (ii) & (iii), we get
Area of ∆APQ = Area of ∆BPR
Hence proved
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