Advertisements
Advertisements
प्रश्न
In ΔABC, E and F are mid-points of sides AB and AC respectively. If BF and CE intersect each other at point O,
prove that the ΔOBC and quadrilateral AEOF are equal in area.
उत्तर
E and F are the midpoints of the sides AB and AC.
Consider the following figure.
Therefore, by midpoint theorem, we have, EF || BC
Triangles BEF and CEF lie on the common base EF and between the parallels, EF and BC
Therefore, Ar.( ΔBEF ) = Ar.( ΔCOF )
⇒ Ar.( ΔBOE ) + Ar.( ΔEOF ) = Ar.( ΔEOF ) + Ar.( ΔCOF )
⇒ Ar.(ΔBOE ) = Ar.( ΔCOF )
Now BF and CE are the medians of the triangle ABC
Medians of the triangle divide it into two equal areas of triangles.
Thus, we have, Ar. (ΔABF) = Ar. (ΔCBF)
Subtracting Ar. ΔBOE on both the sides, we have
Ar. (ΔABF) - Ar. (ΔBOE) = Ar. (ΔCBF) - Ar. (ΔBOE)
Since, Ar. ( ΔBOE ) = Ar. ( ΔCOF ),
Ar. (ΔABF) - Ar. (ΔBOE) = Ar. (ΔCBF) - Ar. (ΔCOF)
Ar. ( quad. AEOF ) = Ar. ( ΔOBC ) , hence proved
APPEARS IN
संबंधित प्रश्न
In the given figure, ABCD is a parallelogram; BC is produced to point X.
Prove that: area ( Δ ABX ) = area (`square`ACXD )
In the following, AC // PS // QR and PQ // DB // SR.
Prove that: Area of quadrilateral PQRS = 2 x Area of the quad. ABCD.
In the following figure, CE is drawn parallel to diagonals DB of the quadrilateral ABCD which meets AB produced at point E.
Prove that ΔADE and quadrilateral ABCD are equal in area.
In the figure given alongside, squares ABDE and AFGC are drawn on the side AB and the hypotenuse AC of the right triangle ABC.
If BH is perpendicular to FG
prove that:
- ΔEAC ≅ ΔBAF
- Area of the square ABDE
- Area of the rectangle ARHF.
ABCD is a parallelogram a line through A cuts DC at point P and BC produced at Q. Prove that triangle BCP is equal in area to triangle DPQ.
Show that:
A diagonal divides a parallelogram into two triangles of equal area.
ABCD is a parallelogram. P and Q are the mid-points of sides AB and AD respectively.
Prove that area of triangle APQ = `1/8` of the area of parallelogram ABCD.
ABCD is a trapezium with AB parallel to DC. A line parallel to AC intersects AB at X and BC at Y.
Prove that the area of ∆ADX = area of ∆ACY.
In the given figure, the diagonals AC and BD intersect at point O. If OB = OD and AB//DC,
show that:
(i) Area (Δ DOC) = Area (Δ AOB).
(ii) Area (Δ DCB) = Area (Δ ACB).
(iii) ABCD is a parallelogram.
Show that:
The ratio of the areas of two triangles on the same base is equal to the ratio of their heights.
