In △ ABC, E is mid-point of the median AD and BE produced meets side AC at point Q. Show that BE : EQ = 3 : 1.

Draw DY || BQ. In △ BCQ and △ DCY, ⇒ ∠BCQ = ∠DCY (Common) ⇒ ∠BQC = ∠DYC (Corresponding angles are equal) ∴ △ BCQ ~ △ DCY (By A.A. axiom) We know that, Corresponding sides of similar triangle are proportional. ..........(1) Since, D is the mid-point of BC. ∴ BC = 2CD Considering L.H.S. of the equation (1), we get : In △ AEQ and △ ADY, ⇒ ∠EAQ = ∠DAY (Common) ⇒ ∠AEQ = ∠ADY (Corresponding angles are equal) ∴ △ AEQ ~ △ ADY (By A.A. axiom) We know that, Corresponding sides of similar triangle are proportional. (Since, E is the mid-point of AD) ............(2) Dividing equation (1) by (2), we get : Hence, proved that BE : EQ = 3 : 1.

P, Q and R are mid-points of sides AB, BC and CD respectively of a rhombus ABCD. Show that PQ is perpendicular to QR.

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The diagonals of a quadrilateral ABCD are perpendicular to each other. Prove that the quadrilateral obtained by joining the mid-points of its adjacent sides is a rectangle.

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In the given figure, M is the mid-point of AB and DE, whereas N is mid-point of BC and DF. Show that : EF = AC.

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In triangle ABC; D and E are mid-points of the sides AB and AC respectively. Through E, a straight line is drawn parallel to AB to meet BC at F. Prove that BDEF is a parallelogram. If AB = 16 cm, AC = 12 cm and BC = 18 cm, find the perimeter of the parallelogram BDEF.

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Mathematics

In △ ABC, E is mid-point of the median AD and BE produced meets side AC at point Q. Show that BE : EQ = 3 : 1.

Mid-point Theorem

Related Questions

P, Q and R are mid-points of sides AB, BC and CD respectively of a rhombus ABCD. Show that PQ is perpendicular to QR.

The diagonals of a quadrilateral ABCD are perpendicular to each other. Prove that the quadrilateral obtained by joining the mid-points of its adjacent sides is a rectangle.

In the given figure, M is the mid-point of AB and DE, whereas N is mid-point of BC and DF. Show that : EF = AC.

In triangle ABC; D and E are mid-points of the sides AB and AC respectively. Through E, a straight line is drawn parallel to AB to meet BC at F. Prove that BDEF is a parallelogram. If AB = 16 cm, AC = 12 cm and BC = 18 cm, find the perimeter of the parallelogram BDEF.