Advertisements
Advertisements
प्रश्न
ΔABC is an isosceles triangle with AB = AC. D, E and F are the mid-points of BC, AB and AC respectively. Prove that the line segment AD is perpendicular to EF and is bisected by it.
Advertisements
उत्तर
Since the segment joining the mid-points of two sides of a triangle is parallel to third side and is half of it,
Therefore,
DE || AB, DE = `(1)/(2)"AB"`
Also,
DF || AC, DF = `(1)/(2)"AC"`
But AB = AC
⇒ `(1)/(2)"AB" = (1)/(2)"AC"`
⇒ DF = DE ........(i)
DE = `(1)/(2)"AB"`
⇒ DE = AF ........(ii)
And DF = `(1)/(2)"AC"`
⇒ DF = AE ........(iii)
From (i), (ii) and (iii)
DE = AE = EF = DF
⇒ DEAF is a rhombus.
⇒ Diagonals AD and EF bisect each other at right angles.
⇒ AD perpendicular to EF and AD is bisected by EF.
APPEARS IN
संबंधित प्रश्न
In below fig. ABCD is a parallelogram and E is the mid-point of side B If DE and AB when produced meet at F, prove that AF = 2AB.
In ∆ABC, E is the mid-point of the median AD, and BE produced meets side AC at point Q.
Show that BE: EQ = 3: 1.
In the figure, give below, 2AD = AB, P is mid-point of AB, Q is mid-point of DR and PR // BS. Prove that:
(i) AQ // BS
(ii) DS = 3 Rs.
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.
In ΔABC, AB = 12 cm and AC = 9 cm. If M is the mid-point of AB and a straight line through M parallel to AC cuts BC in N, what is the length of MN?
Prove that the figure obtained by joining the mid-points of the adjacent sides of a rectangle is a rhombus.
In parallelogram ABCD, P is the mid-point of DC. Q is a point on AC such that CQ = `(1)/(4)"AC"`. PQ produced meets BC at R. Prove that
(i) R is the mid-point of BC, and
(ii) PR = `(1)/(2)"DB"`.
ABCD is a parallelogram.E is the mid-point of CD and P is a point on AC such that PC = `(1)/(4)"AC"`. EP produced meets BC at F. Prove that: 2EF = BD.
The figure obtained by joining the mid-points of the sides of a rhombus, taken in order, is ______.
In ∆ABC, AB = 5 cm, BC = 8 cm and CA = 7 cm. If D and E are respectively the mid-points of AB and BC, determine the length of DE.
