Advertisements
Advertisements
Question
ABCD is parallelogram and E is a point on BC. If the diagonal BD intersects AE at F, prove that AF × FB = EF × FD.
Solution
We have:
∠𝐴𝐹𝐷= ∠𝐸𝐹𝐵 (𝑉𝑒𝑟𝑡𝑖𝑐𝑎𝑙𝑙𝑦 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑎𝑛𝑔𝑙𝑒𝑠)
∵ DA || BC
∴ ∠𝐷𝐴𝐹= ∠𝐵𝐸𝐹 (𝐴𝑙𝑡𝑒𝑟𝑛𝑎𝑡𝑒 𝑎𝑛𝑔𝑙𝑒𝑠)
Δ DAF ~ Δ BEF (AA similarity theorem)
⟹ 𝐴𝐹𝐸𝐹=𝐹𝐷𝐹𝐵
Or, AF × FB = FD × EF
This completes the proof
APPEARS IN
RELATED QUESTIONS
In an isosceles ∆ABC, the base AB is produced both ways in P and Q such that AP × BQ = AC2 and CE are the altitudes. Prove that ∆ACP ~ ∆BCQ.
In figure, ABCD is a trapezium with AB || DC. If ∆AED is similar to ∆BEC, prove that AD = BC.
E and F are points on the sides PQ and PR, respectively, of a ΔPQR. For the following case, state whether EF || QR.
PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.36 cm
The perimeter of two similar triangles are 30 cm and 24 cm. If one side of the first triangle is 12 cm, determine the corresponding side of the second triangle.
The given figure shows a triangle PQR in which XY is parallel to QR. If PX : XQ = 1 : 3 and QR = 9 cm, find the length of XY.
Further, if the area of ΔPXY = x cm2; find, in terms of x the area of :
- triangle PQR.
- trapezium XQRY.
ABCD is a quadrilateral in which AD = BC. If P, Q, R, S be the midpoints of AB, AC, CD and BD respectively, show that PQRS is a rhombus.
The areas of two similar triangles are `64cm^2` and `100cm^2` respectively. If a median of the smaller triangle is 5.6cm, find the corresponding median of the other.
In the given figure, DE║BC and DE: BC = 3:5. Calculate the ratio of the areas of ΔADE and the trapezium BCED.
State the AA-similarity criterion
If in ∆DEF and ∆PQR, ∠D ≅ ∠Q, ∠R ≅ ∠E then which of the following statements is false?
O is any point in the interior of ΔABC. Bisectors of ∠AOB, ∠BOC and ∠AOC intersect side AB, side BC, side AC in
F, D and E respectively.
Prove that
BF × AE × CD = AF × CE × BD
In Δ ABC , MN || BC .
If BC = 14 cm and MN = 6 cm , find `("Ar" triangle "AMN")/("Ar" . ("trapezium MBCN"))`
A ship is 400m laig and 100m wide. The length of its model is 20 cm. find the surface area of the deck of the model.
In the given figure, PQ || AB; CQ = 4.8 cm QB = 3.6 cm and AB = 6.3 cm. Find : PQ
In the following figure, M is mid-point of BC of a parallelogram ABCD. DM intersects the diagonal AC at P and AB produced at E. Prove that : PE = 2 PD
In fig.DE || BC ,AD = 1 cm and BD = 2 cm. what is the ratio of the ar(ΔABC) to the ar (ΔADE)?
Construct a triangle ABC with side BC = 6 cm, ∠B = 45°, ∠A = 105°. Then construct another triangle whose sides are `(3)/(4)` times the corresponding sides of the ΔABC.
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 figure, PQ is parallel to BC, AP : AB = 2 : 7. If QC = 0 and BC = 21,
Find
(i) AQ
(ii) PQ
In ΔABC, point D divides AB in the ratio 5:7, Find: `"AE"/"EC"`
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")`
In ΔABC, DE || BC such that AD =1.5 cm, DB = 3 cm and AE = 1 cm. Find AC.
Find the scale factor in each of the following and state the type of size transformation:
Model area = 75cm2, Actual area = 3cm2
In the adjacent figure, ∆ACB ~ ∆APQ. If BC = 8 cm, PQ = 4 cm, BA = 6.5 cm and AP = 2.8 cm, find CA and AQ
Given ΔABC ~ ΔDEF, if ∠A = 45° and ∠E = 35° then ∠B = ?
In fig. BP ⊥ AC, CQ ⊥ AB, A−P−C, and A−Q−B then show that ΔAPB and ΔAQC are similar.
In ΔAPB and ΔAQC
∠APB = [ ]° ......(i)
∠AQC = [ ]° ......(ii)
∠APB ≅ ∠AQC .....[From (i) and (ii)]
∠PAB ≅ ∠QAC .....[______]
ΔAPB ~ ΔAQC .....[______]
∆ABC ~ ∆PQR. If AM and PN are altitudes of ΔABC and ∆PQR respectively and AB2 : PQ2 = 4 : 9, then AM : PN = ______.
The ratio of the corresponding altitudes of two similar triangles is `3/5`. Is it correct to say that ratio of their areas is `6/5`? Why?
