Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
Construction: Draw DX || BQ
In ΔBCQ and ΔDCX,
∠BCQ = ∠DCX ...(Common)
∠BQC = ∠DXC ...(Corresponding angles)
So, ΔBCQ ∼ ΔDCX ....(AA Similarity criterion)
⇒ `"BQ"/"DX" = "BC"/"DC" = "CQ"/"CX"` ...(Corresponding sides are proportional.)
⇒ `"BQ"/"DX" = "2CD"/"CD"` ...(D is the mid-point of BC)
⇒ `"BQ"/"DX" = 2` ...(i)
Similarly, ΔAEQ ∼ ΔADX,
⇒ `"EQ"/"DX" = "AE"/"ED" = 1/2` ...(E is the mid-point of AD)
That is `"EQ"/"DX" = 1/2` ...(ii)
Dividing (i) by (ii), We get
⇒ `"BQ"/"EQ" = 4`
⇒ BE + EQ = 4EQ
⇒ BE = 3EQ
⇒ `"BQ"/"EQ" = 3/1`
APPEARS IN
संबंधित प्रश्न
In a parallelogram ABCD, E and F are the mid-points of sides AB and CD respectively (see the given figure). Show that the line segments AF and EC trisect the diagonal BD.
Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.
In a triangle ∠ABC, ∠A = 50°, ∠B = 60° and ∠C = 70°. Find the measures of the angles of
the triangle formed by joining the mid-points of the sides of this triangle.
Fill in the blank to make the following statement correct:
The triangle formed by joining the mid-points of the sides of a right triangle is
The following figure shows a trapezium ABCD in which AB // DC. P is the mid-point of AD and PR // AB. Prove that:
PR = `[1]/[2]` ( AB + CD)
If the quadrilateral formed by joining the mid-points of the adjacent sides of quadrilateral ABCD is a rectangle,
show that the diagonals AC and BD intersect at the right angle.
The diagonals AC and BD of a quadrilateral ABCD intersect at right angles. Prove that the quadrilateral formed by joining the midpoints of quadrilateral ABCD is a rectangle.
In the given figure, PS = 3RS. M is the midpoint of QR. If TR || MN || QP, then prove that:
ST = `(1)/(3)"LS"`
The figure obtained by joining the mid-points of the sides of a rhombus, taken in order, is ______.
D, E and F are the mid-points of the sides BC, CA and AB, respectively of an equilateral triangle ABC. Show that ∆DEF is also an equilateral triangle.
