Advertisements
Advertisements
Question
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)
Solution
Given:
Seg PQ || seg DE
seg QR || seg EF
To Prove: seg PR || seg DF
Proof :
In ΔXDE, PQ || DE ... Given
∴ `"XP"/underline("PD") = underline("XQ")/"QE"` ...(I)(Basic proportionality theorem)
In ΔXEF, QR || EF ... Given
∴ `underline("XR")/underline("RF") = underline("XQ")/underline("QE")` ...(II)(Basic proportionality theorem)
∴ `underline("XP")/underline("PD") = underline("XR")/underline("RF")` ...from (I) and (II)
∴ seg PR || seg DF ...(converse of basic proportionality theorem)
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 the given figure ΔABC and ΔAMP are right angled at B and M respectively. Given AC = 10 cm, AP = 15 cm and PM = 12 cm.
1) Prove ΔABC ~ ΔAMP
2) Find AB and BC.
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.
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:
The corresponding sides of two similar triangles ABC and DEF are BC = 9.1cm and EF = 6.5cm. If the perimeter of ΔDEF is 25cm, find the perimeter of ΔABC.
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.
In ∆ABC and ∆DEF ∠B = ∠E, ∠F = ∠C and AB = 3DE then which of the statements regarding the two triangles is true ?
In the given figure, seg XY || seg BC, then which of the following statements is true?
In the given figure, ∠PQR = ∠PST = 90° ,PQ = 5cm and PS = 2cm.
(i) Prove that ΔPQR ~ ΔPST.
(ii) Find Area of ΔPQR : Area of quadrilateral SRQT.
In Δ ABC , DE is parallel to BC and `"AD"/"DB" = 2/7` IF AC = 5 .6 , find AE.
In Δ ABC , MN || BC .
If BC = 14 cm and MN = 6 cm , find `("Ar" triangle "AMN")/("Ar" . ("trapezium MBCN"))`
On a map drawn to a scale of 1 : 25000, a triangular plot of a land is marked as ABC with AB= 6cm, BC = 8cm and ∠ ABC = 90° . Calculate the actual length of AB in km and the actual area of the plot in km2 .
The dimensions of the model of a multistorey building are 1.2 m × 75 cm × 2 m. If the scale factor is 1 : 30; find the actual dimensions of the building.
Find the scale factor in each of the following and state the type of size transformation:
Image length = 6cm, Actual length = 4cm.
A plot of land of area 20km2 is represented on the map with a scale factor of 1:200000. Find: The area on the map that represented the plot of land.
A model of cargo tuck is made to a scale of 1:40. The length of the model is 15cm. Calculate: The volume of the model if the volume of the truck is 6m3
If in triangles ABC and EDF, `"AB"/"DE" = "BC"/"FD"` then they will be similar, when
Two similar triangles will always have ________ angles
In figure, if AD = 6 cm, DB = 9 cm, AE = 8 cm and EC = 12 cm and ∠ADE = 48°. Find ∠ABC.
