Advertisements
Advertisements
Question
The diagonals of a quadrilateral ABCD intersect each other at the point O such that `("AO")/("BO") = ("CO")/("DO")`. Show that ABCD is a trapezium.
Solution
Let us consider the following figure for the given question.
Draw a line OE || AB
In ΔABD, OE || AB
By using basic proportionality theorem, we obtain
`("AE")/("ED") = ("BO")/("OD")` ...(1)
However, it is given that
`("AO")/("OC") = ("OB")/("OD")` ...(2)
From equation (1) and (2) we obtain
`("AE")/("ED") = ("AO")/("OC")`
⇒ EO || DC ...[By the converse of basic proportionality theorem]
⇒ AB || OE || DC
⇒ AB || CD
∴ ABCD is a trapezium.
APPEARS IN
RELATED QUESTIONS
In figure, ∆ACB ~ ∆APQ. If BC = 8 cm, PQ = 4 cm, BA = 6.5 cm, AP = 2.8 cm, find CA and AQ.
In figure, `\frac{AO}{OC}=\frac{BO}{OD}=\frac{1}{2}` and AB = 5 cm. Find the value of DC.
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.
Prove that, in a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.
State, true or false:
Two congruent polygons are necessarily similar.
In the given figure, ∆ABC and ∆AMP are right angled at B and M respectively.
Given AC = 10 cm, AP = 15 cm and PM = 12 cm.
- Prove that: ∆ABC ~ ∆AMP
- Find: AB and BC.
In each of the given pairs of triangles, find which pair of triangles are similar. State the similarity criterion and write the similarity relation in symbolic form:
In each of the given pairs of triangles, find which pair of triangles are similar. State the similarity criterion and write the similarity relation in symbolic form:
In the given figure, ∠CAB = 90° and AD⊥BC. Show that ΔBDA ~ ΔBAC. If AC = 75cm, AB = 1m and BC = 1.25m, find AD.
ΔABC~ΔPQR and ar(ΔABC) = 4, ar(ΔPQR) . If BC = 12cm, find QR.
State the AA-similarity criterion
In ∆ABC, ray BD bisects ∠ABC and ray CE bisects ∠ACB. If seg AB ≅ seg AC then prove that ED || BC.
If Δ ABC , MN || BC .
If AN : AC= 5 : 8, find ar(Δ AMN) : ar(Δ ABC)
In Δ PQR, MN is drawn parallel to QR. If PM = x, MQ = (x-2), PN = (x+2) and NR = (x-1), find the value of x.
ABCD and PQRS are similar figures. AB= 12cm, BC=x cm, CD= 15 cm, AD= 10 cm, PQ= 8 cm, QR = 5 cm, RS = m cm and PS = n cm .Find the values of x, m and n.
Δ ABC ∼ Δ PQR such that AB= 1.5 cm and PQ=2. 1 cm. Find the ratio of areas of Δ ABC and ΔPQR.
A model of a ship is made with a scale factor of 1 : 500. Find
The volume of the ship, if the volume of its model is 200 cm3.
The scale of a map is 1 : 200000. A plot of land of area 20km2 is to be represented on the map. Find
The area on the map that represents the plot of land.
If ΔABC ~ ΔPQR and ∠A = 60°, then ∠P = ?
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.
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.
In the following figure, point D divides AB in the ratio 3 : 5. Find : `(AE)/(AC)`
In the following figure, point D divides AB in the ratio 3 : 5. Find :
BC = 4.8 cm, find the length of DE.
A line PQ is drawn parallel to the side BC of ΔABC which cuts side AB at P and side AC at Q. If AB = 9.0 cm, CA = 6.0 cm and AQ = 4.2 cm, find the length of AP.
In the figure, given below, PQR is a right-angle triangle right angled at Q. XY is parallel to QR, PQ = 6 cm, PY = 4 cm and PX : XQ = 1 : 2. Calculate the lengths of PR and QR.
Equilateral triangles are drawn on the sides of a right angled triangle. Show that the area of the triangle on the hypotenuse is equal to the sum of the areas of triangles on the other two sides.
In the adjoining figure. BC is parallel to DE, area of ΔABC = 25 sq cm, area of trapezium BCED = 24 sq cm, DE = 14 cm. Calculate the length of BC.
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.
If ΔABC, D and E are points on AB and AC. Show that DE || BC for each of the following case or not:
AB = 10.8cm, BD = 4.5cm, AC = 4.8cm, and AE = 2.8cm
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")`
AM and DN are the altitudes of two similar triangles ABC and DEF. Prove that: AM : DN = AB : DE.
Through the vertex S of a parallelogram PQRS, a line is drawn to intersect the sides Qp and QR produced at M and N respectively. Prove that `"SP"/"PM" = "MQ"/"QN" = "MR"/"SR"`
In the given figure, PB is the bisector of ABC and ABC =ACB. Prove that:
a. BC x AP = PC x AB
b. AB:AC = BP: BC
The areas of two similar triangles are 16cm2 and 9cm2 respectively. If the altitude of the smaller triangle is 1.8cm, find the length of the altitude corresponding to the larger triangle.
Find the scale factor in each of the following and state the type of size transformation:
Actual length = 12cm, Image length = 15cm.
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 covered by 6cm on the map
A model of a ship is made to a scale of 1:500. Find: The area other deck o the ship, if the area of the deck of its model is m2
Check whether the triangles are similar and find the value of x
A vertical stick of length 6 m casts a shadow 400 cm long on the ground and at the same time a tower casts a shadow 28 m long. Using similarity, find the height of the tower
In the adjacent figure, ∆ABC is right angled at C and DE ⊥ AB. Prove that ∆ABC ~ ∆ADE and hence find the lengths of AE and DE
If figure OPRQ is a square and ∠MLN = 90°. Prove that ∆LOP ~ ∆QMO
Two vertical poles of heights 6 m and 3 m are erected above a horizontal ground AC. Find the value of y
Construct a triangle similar to a given triangle ABC with its sides equal to `6/5` of the corresponding sides of the triangle ABC (scale factor `6/5 > 1`)
In ∆LMN, ∠L = 60°, ∠M = 50°. If ∆LMN ~ ∆PQR then the value of ∠R is
D is the mid point of side BC and AE ⊥ BC. If BC = a, AC = b, AB = c, ED = x, AD = p and AE = h, prove that b2 + c2 = `2"p"^2 + "a"^2/2`
A man whose eye-level is 2 m above the ground wishes to find the height of a tree. He places a mirror horizontally on the ground 20 m from the tree and finds that if he stands at a point C which is 4 m from the mirror B, he can see the reflection of the top of the tree. How height is the tree?
In the given figure YH || TE. Prove that ΔWHY ~ ΔWET and also find HE and TE
In the given figure, if ΔEAT ~ ΔBUN, find the measure of all angles.
Are triangles in figure similar? If yes, then write the test of similarity.
In given fig., quadrilateral PQRS, side PQ || side SR, AR = 5 AP, then prove that, SR = 5PQ
In Quadrilateral ABCD, side AD || BC, diagonal AC and BD intersect in point P, then prove that `"AP"/"PD" = "PC"/"BP"`
ΔDEF ~ ΔABC. If DE : AB = 2 : 3 and ar ΔDEF is equal to 44 square units then ar (ΔABC) (square unit) is ______.
In a square of side 10 cm, its diagonal = ______.
It is given that ΔABC ~ ΔPQR, with `(BC)/(QR) = 1/3`. Then, `(ar(PRQ))/(ar(BCA))` is equal to ______.
In figure, if AD = 6cm, DB = 9cm, AE = 8cm and EC = 12cm and ∠ADE = 48°. Find ∠ABC.
