Advertisements
Advertisements
Question
In the given figure, DE║BC and DE: BC = 3:5. Calculate the ratio of the areas of ΔADE and the trapezium BCED.
Solution
It is given that DE || BC.
∴ ∠𝐴𝐷𝐸= ∠𝐴𝐵𝐶 (𝐶𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑖𝑛𝑔 𝑎𝑛𝑔𝑙𝑒𝑠)
∠𝐴𝐸𝐷= ∠𝐴𝐶𝐵 (𝐶𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑖𝑛𝑔 𝑎𝑛𝑔𝑙𝑒𝑠)
Applying AA similarity theorem, we can conclude that Δ ADE ~ ΔABC.
∴`( ar(ΔABC))/(ar(ΔADE))=(BC)^2/(DE)^2`
Subtracting 1 from both sides, we get:
`(ar(ΔABC))/(ar(ΔADE))-1=5^2/3^2-1`
⇒`( ar(ΔABC)-ar(ΔADE))/(ar(ΔADE))=(25-9)/9`
⇒ `(ar(BCED))/(ar(ΔADE))=16/9`
Or, `(ar(ΔADE))/(ar(BCED))=9/16`
APPEARS IN
RELATED QUESTIONS
In figure, find ∠L
E and F are points on the sides PQ and PR, respectively, of a ΔPQR. For the following case, state whether EF || QR:
PE = 4 cm, QE = 4.5 cm, PF = 8 cm and RF = 9 cm
State, true or false:
Two congruent polygons are necessarily similar.
Given: ∠GHE = ∠DFE = 90°,
DH = 8, DF = 12,
DG = 3x – 1 and DE = 4x + 2.
Find: the lengths of segments DG and DE.
In the given figure, ∠ABC = 90° and BD⊥AC. If AB = 5.7cm, BD = 3.8cm and CD = 5.4cm, find BC.
ΔABC~ΔPQR and ar(ΔABC) = 4, ar(ΔPQR) . If BC = 12cm, find QR.
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.
State the AA-similarity criterion
State the SAS-similarity criterion
In the given figure, ∠ABC = 75°, ∠EDC = 75° state which two triangles are similar and by which test? Also write the similarity of these two triangles by a proper one to one correspondence.
Are the triangles in the given figure similar? If yes, by which test?
If Δ ABC , MN || BC .
If AN : AC= 5 : 8, find ar(Δ AMN) : ar(Δ ABC)
In the figure , ABCD is a quadrilateral . F is a point on AD such that AF = 2.1 cm and FD = 4.9 cm . E and G are points on AC and AB respectively such that EF || CD and GE || BC . Find `("Ar" triangle "BCD")/("Ar" triangle "GEF")`
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.
A model of a ship is made with a scale factor of 1 : 500. Find
The length of the ship, if the model length is 60 cm.
A triangle ABC has been enlarged by scale factor m = 2.5 to the triangle A' B' C'. Calculate : the length of AB, if A' B' = 6 cm.
In the following figure, point D divides AB in the ratio 3 : 5. Find :
BC = 4.8 cm, find the length of DE.
Find the area of the triangle ABC with the coordinates of A as (1, −4) and the coordinates of the mid-points of sides AB and AC respectively are (2, −1) and (0, −1).
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 = 4x - 3, AE = 8x - 7, BD = 3x - 1 and CE = 5x - 3,Find x.
A plot of land of area 20km2 is represented on the map with a scale factor of 1:200000. Find: The ground area in km2 that is represented by 2cm2 on the map.
A map is drawn to scale of 1:20000. Find: The distance on the map representing 4km
If ∆ABC is an isosceles triangle with ∠C = 90° and AC = 5 cm, then AB is
If BD ⊥ AC and CE ⊥ AB, prove that ∆AEC ~ ∆ADB
In the figure, which of the following statements is true?
Which of the following is not a test of similarity?
There are two poles having heights 8 m and 4 m on plane ground as shown in fig. Because of sunlight shadows of smaller pole is 6m long, then find the length of shadow of longer pole.
In given fig., quadrilateral PQRS, side PQ || side SR, AR = 5 AP, then prove that, SR = 5PQ
In ΔPQR, S and T are points on PQ and PR respectively. `(PS)/(SQ) = (PT)/(TR)` and ∠PST = ∠PRQ. Prove that PQR is an isosceles triangle.
