Advertisements
Advertisements
प्रश्न
Using Converse of basic proportionality theorem, prove that the line joining the mid-points of any two sides of a triangle is parallel to the third side. (Recall that you have done it in Class IX).
उत्तर
Consider the given figure in which PQ is a line segment joining the mid-points P and Q of line AB and AC respectively.
i.e., AP = PB and AQ = QC
It can be observed that
`("AP")/("PB") = 1/1`
and `("AQ")/("QC") = 1/1`
∴ `("AP")/("PB") = ("AQ")/("QC")`
Hence, by using basic proportionality theorem, we obtain
PQ || BC
APPEARS IN
संबंधित प्रश्न
In figure, considering triangles BEP and CPD, prove that BP × PD = EP × PC.
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
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).
In ΔPQR, MN is parallel to QR and `(PM)/(MQ) = 2/3`
1) Find `(MN)/(QR)`
2) Prove that ΔOMN and ΔORQ are similar.
3) Find, Area of ΔOMN : Area of ΔORQ
In ΔABC, ∠ABC = ∠DAC, AB = 8 cm, AC = 4 cm and AD = 5 cm.
- Prove that ΔACD is similar to ΔBCA.
- Find BC and CD.
- Find area of ΔACD : area of ΔABC.
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.
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:
P and Q are points on the sides AB and AC respectively of a ΔABC. If AP = 2cm, PB = 4cm, AQ = 3cm and QC = 6cm, show that BC = 3PQ.
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.
State the AA-similarity criterion
In ∆ABC and ∆DEF ∠B = ∠E, ∠F = ∠C and AB = 3DE then which of the statements regarding the two triangles is true ?
Δ ABC - Δ XYZ. If area of Δ ABC is 9cm2 and area of Δ XYZ is 16cm2 and if BC= 2.1cm, find the length of YZ.
If Δ ABC , MN || BC .
If AN : AC= 5 : 8, find ar(Δ AMN) : ar(Δ ABC)
In Δ ABC , MN || BC .
If BC = 14 cm and MN = 6 cm , find `("Ar" triangle "AMN")/("Ar" . ("trapezium MBCN"))`
Figure shows Δ KLM , P an T on KL and KM respectively such that∠ KLM =∠ KTP.
If `"KL"/"KT" = 9/5` , find `("Ar" triangle "KLM")/("Ar" triangle "KTP")`.
In figure , DEF is a right -angled triangle with ∠ E = 90 °.FE is produced to G and GH is drawn perpendicular to DE = 8 cm , DH = 8 cm ,DH = 6 cm and HF = 4 cm , find `("Ar" triangle "DEF")/("Ar" triangle "GHF")`
In MBC, DE is drawn parallel to BC. If AD: DB=2:3, DE =6cm and AE =3.6cm, find BC and AC.
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 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 LMN has been reduced by scale factor 0.8 to the triangle L' M' N'. Calculate: the length of LM, if L' M' = 5.4 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 angle ABC = 90°.
Calculate : the area of the plot in sq. km.
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.
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.
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 the given figure, AB and DE are perpendicular to BC.
- Prove that ΔABC ∼ ΔDEC
- If AB = 6 cm, DE = 4 cm and AC = 15 cm. Calculate CD.
- Find the ratio of the area of a ΔABC : area of ΔDEC.
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.
If ΔABC, D and E are points on AB and AC. Show that DE || BC for each of the following case or not:
AB = 5.6cm, AD = 1.4cm, AC = 7.2cm, and AE = 1.8cm
In ΔABC, point D divides AB in the ratio 5:7, Find: DE, If BC = 4.8cm
Two figures are similar. If the ratio of their perimeters is 8:16. What will be the ratio of the corresponding sides?
The scale of a map is 1 : 50000. The area of a city is 40 sq km which is to be represented on the map. Find: The area of land represented 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 diagonal distance of the plot in km
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
Two triangles QPR and QSR, right angled at P and S respectively are drawn on the same base QR and on the same side of QR. If PR and SQ intersect at T, prove that PT × TR = ST × TQ
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 ∆QMO ~ ∆RPN
Construct a triangle similar to a given triangle PQR with its sides equal to `2/3` of the corresponding sides of the triangle PQR (scale factor `2/3 < 1`)
PR = 26 cm, QR = 24 cm, ∠PAQ = 90°, PA = 6 cm and QA = 8 cm. Find ∠PQR
In the given figure AB || CD || EF. If AB = 6 cm, CD = x cm, EF = 4 cm, BD = 5 cm and DE = y cm. Find x and y.
If ∆ABC – ∆PQR in which ∠A = 53° and ∠Q = 77°, then ∠R is
In the figure, if ∠FEG ≡ ∠1 then, prove that DG2 = DE.DF
State whether the following triangles are similar or not: If yes, then write the test of similarity.
∠P = 35°, ∠X = 35° and ∠Q = 60°, ∠Y = 60°
Observe the figure and complete the following activity
In fig, ∠B = 75°, ∠D = 75°
∠B ≅ [ ______ ] ...[each of 75°]
∠C ≅ ∠C ...[ ______ ]
ΔABC ~ Δ [ ______ ] ...[ ______ similarity test]
ΔABC ~ ΔPQR, A(ΔABC) = 80 sq.cm, A(ΔPQR) = 125 sq.cm, then complete `("A"(Δ"ABC"))/("A"(Δ"PQR")) = 80/125 = (["______"])/(["______"])`, hence `"AB"/"PQ" = (["______"])/(["______"])`
ΔABP ~ ΔDEF and A(ΔABP) : A(ΔDEF) = 144:81, then AB : DE = ?
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.
Two triangles are similar. Smaller triangle’s sides are 4 cm, 5 cm, 6 cm. Perimeter of larger triangle is 90 cm then find the sides of larger triangle.
In the given figure the value of x is
In ∠BAC = 90° and AD ⊥ BC. A then ______.
In ΔABC, PQ || BC. If PB = 6 cm, AP = 4 cm, AQ = 8 cm, find the length of AC.
In figure, if AD = 6 cm, DB = 9 cm, AE = 8 cm and EC = 12 cm and ∠ADE = 48°. Find ∠ABC.
In the adjoining diagram the length of PR is ______.
