Advertisements
Advertisements
प्रश्न
The diagonal BD of a parallelogram ABCD intersects the segment AE at the point F, where E is any point on the side BC. Prove that DF × EF = FB × FA
उत्तर
In ∆AFD and ∆BFE, we have
∠1 = ∠2 [Vertically opposite angles]
∠3 = ∠4 [Alternate angles]
AD || BC
So, by AA-criterion of similarity, we have
∆FBE ~ ∆FDA
⇒ `(FB)/(FD) = (FD)/(FA)`
⇒ `(FB)/(DF)= (EF)/(FA)`
⇒ DF × EF = FB × FA
APPEARS IN
संबंधित प्रश्न
Prove that the angle bisector of a triangle divides the side opposite to the angle in the ratio of the remaining sides.
In figure, `\frac{AO}{OC}=\frac{BO}{OD}=\frac{1}{2}` and AB = 5 cm. Find the value of DC.
Using Basic proportionality theorem, prove that a line drawn through the mid-points of one side of a triangle parallel to another side bisects the third side. (Recall that you have proved it in Class IX).
Show that ΔABC, where A(–2, 0), B(2, 0), C(0, 2) and ΔPQR where P(–4, 0), Q(4, 0), R(0, 2) are similar triangles
A line PQ is drawn parallel to the base BC of ∆ABC which meets sides AB and AC at points P and Q respectively. If AP = `1/3` PB; find the value of:
- `"Area of ΔABC"/"Area of ΔAPQ"`
- `"Area of ΔAPQ"/"Area of trapezium PBCQ"`
In the given figure, BC is parallel to DE. Area of triangle ABC = 25 cm2, Area of trapezium BCED = 24 cm2 and DE = 14 cm. Calculate the length of BC. Also, find the area of triangle BCD.
In Figure 3, ABCD is a trapezium with AB || DC, AB = 18 cm, DC = 32 cm and the distance between AB and DC is 14 cm. If arcs of equal radii 7 cm have been drawn, with centres A,B, C and D, then find the area of the shaded region.
In the given figure, X is any point in the interior of triangle. Point X is joined to vertices of triangle. Seg PQ || seg DE, seg QR || seg EF. Fill in the blanks to prove that, seg PR || seg DF.
Proof : In ΔXDE, PQ || DE ...`square`
∴ `"XP"/square = square/"QE"` ...(I) (Basic proportionality theorem)
In ΔXEF, QR || EF ...`square`
∴ `square/square = square/square ..."(II)" square`
∴ `square/square = square/square` ...from (I) and (II)
∴ seg PR || seg DF ...(converse of basic proportionality theorem)
A triangle ABC has been enlarged by scale factor m = 2.5 to the triangle A' B' C'. Calculate : the length of C' A' if CA = 4 cm.
A triangle ABC is enlarged, about the point O as centre of enlargement, and the scale factor is 3. Find : BC, if B' C' = 15 cm.
On a map drawn to a scale of 1 : 2,50,000; a triangular plot of land has the following measurements : AB = 3 cm, BC = 4 cm and ∠ABC = 90°.
Calculate : the actual lengths of AB and BC in km.
In the given figure, PQ || AB; CQ = 4.8 cm QB = 3.6 cm and AB = 6.3 cm. Find :
- `(CP)/(PA)`
- PQ
- If AP = x, then the value of AC in terms of x.
In the given figure, PQ || AB; CQ = 4.8 cm QB = 3.6 cm and AB = 6.3 cm. Find : If AP = x, then the value of AC in terms of x.
In the given figure, ΔABC ~ ΔADE. If AE : EC = 4 : 7 and DE = 6.6 cm, find BC. If 'x' be the length of the perpendicular from A to DE, find the length of perpendicular from A to BC in terms of 'x'.
In fig.DE || BC ,AD = 1 cm and BD = 2 cm. what is the ratio of the ar(ΔABC) to the ar (ΔADE)?
Prove that the area of the triangle BCE described on one side BC of a square ABCD as base is one half of the area of similar triangle ACF described on the diagonal AC as base.
In ΔABC, D and E are the mid-point on AB and AC such that DE || BC.
If AD = 4, AE = 8, DB = x - 4 and EC = 3x - 19, find x.
In the figure, PR || SQ. If PR = 10cm, PT = 5cm, TQ = 6cm and ST = 9cm, calculate RT and SQ.
In ΔABC, BP and CQ are altitudes from B and C on AC and AB respectively. BP and CQ intersect at O. Prove that
(i) PC x OQ = QB x OP
(ii) `"OC"^2/"OB"^2 = ("PC" xx "PO")/("QB" xx "QO")`
ΔABC has been reduced by a scale factor 0.6 to ΔA'B'C'/ Calculate:Length of B' C', if BC = 8cm
A map is drawn to scale of 1:20000. Find: The distance covered by 6cm on the map
On a map drawn to a scale of 1:25000, a rectangular plot of land has sides 12cm x 16cm. Calculate: The area of the plot in sq km
In ∆LMN, ∠L = 60°, ∠M = 50°. If ∆LMN ~ ∆PQR then the value of ∠R is
Two similar triangles will always have ________ angles
Write the test of similarity for triangles given in figure.
In the given figure, if ABCD is a trapezium in which AB || CD || EF, then prove that `(AE)/(ED) = (BF)/(FC)`.
In ΔABC, DE || BC (as shown in the figure), If AD = 4 cm, AB = 9 cm and AC = 13.5 cm, then the length of EC is ______.
