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Similarity

In the adjoining figure, ABC is a triangle. DE is parallel to BC and AD/DB = 3/2,
(i) Determine the ratios AD/AB, DE/BC0
(ii) Prove that ∆DEF is similar to ∆CBF. Hence, find EF/FB.
(iii) What is the ratio of the areas of ∆DEF and ∆CBF?

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- (i) We have to find the ratios AD/AB, DE/BC, From the question it is given that, AD/DB = 3/2 Then, DB/AD = 2/3 Now add 1 for both LHS and RHS we get, (DB/AD) + 1 = (2/3) + 1 (DB + AD)/AD...

In the given figure, CM and RN are respectively the medians of ∆ABC and ∆PQR. If ∆ABC ~ ∆PQR, prove that:
(i) ∆AMC ~ ∆PQR
(ii) CM/RN = AB/PQ
(iii) ∆CMB ~ ∆RNQ

CBSE, Class 10, Maths, ML Aggarwal, Similarity

Solution:- From the given figure it is given that, CM and RN are respectively the medians of ∆ABC and ∆PQR. (i) We have to prove that, ∆AMC ~ ∆PQR Consider the ∆ABC and ∆PQR As ∆ABC ~ ∆PQR ∠A = ∠P,...

State which pairs of triangles in the figure given below are similar. Write the similarity rule used and also write the pairs of similar triangles in symbolic form (all lengths of sides are in cm):

CBSE, Class 10, Exercise 1, maths, Maths, ML Aggarwal, Similarity

Solution:- (i) From the ΔABC and ΔPQR AB/PQ = 3.2/4 = 32/40 Divide both numerator and denominator by 8 we get, = 4/5 AC/PR = 3.6/4.5 = 36/45 Divide both numerator and denominator by 9 we get, = 4/5...

10. $\vartriangle XYZ\sim \vartriangle PQR$ , AD and PS are altitudes from X and P on sides YZ and QR respectively. If $XD:PS=4:9$ , find the ratio of the areas of $\vartriangle XYZ$ and $\vartriangle PQR$ .

Class 10, Frank, ICSE, Maths, Similarity

Given that, $\vartriangle XYZ\sim \vartriangle PQR$ and $XD:PS=4:9$ As we know that, the ratio of areas of two similar triangles is equal to the ratio of square of their corresponding sides. Area of...

15. In $\vartriangle XYZ$ , $DE||YZ$ ; DZ and EY intersect at F, if $DE/YZ=2/7$ , find area of $\vartriangle FDE$ /area of $\vartriangle FYZ$

Class 10, Frank, ICSE, Maths, Similarity

Given that, $DE/YZ=2/7$, $DE||YZ$ We Consider the $\vartriangle FDE$ and $\vartriangle FZY$ $\angle FDE=\angle FZY$ … [because interior alternate angles are equal] $\angle XNM=\angle XZY$ … [because...

14. in $\vartriangle XYZ$ , $MN||YZ$ (a) If $XN:YZ=5:8$ , find the ratio of area of $\vartriangle (b) If$ XY/XM=9/4 $, find area of trapezium MBCN/area of$ \vartriangle XYZ$.

Class 10, Frank, ICSE, Maths, Similarity

It is given that, $XN:XZ=5:8$ Now, we have to find the ratio of area of $\vartriangleXMN:are{{a}_{{}}}o{{f}_{{}}}\vartriangle XYZ$ Consider the ΔXMN$\vartriangle XMN$ and ΔXYZ$\vartriangle XYZ$...

13. Prove that the area of $\vartriangle XYE$ described on one side XY of a square WXYZ is one half the area of the similar $\vartriangle WYF$ described on the diagonal WY.

Class 10, Frank, ICSE, Maths, Similarity

We Consider right angle triangle WXY, Pythagoras theorem, $W{{Y}^{2}}=W{{X}^{2}}+X{{Y}^{2}}$ $W{{Y}^{2}}=2X{{Y}^{2}}$… [because WX = XY] It is given that, $\vartriangle XYE\sim \vartriangle WYF$ As...

12. . If area of $\vartriangle PQR$ is $9c{{m}^{2}}$ and area of $\vartriangle XYZ$ is $16c{{m}^{2}}$ and if $QR=2.1cm$ , find the length of YZ.

Class 10, Frank, ICSE, Maths, Similarity

Given that, $\vartriangle PQR\sim \vartriangle XYZ$, area of $\vartriangle PQR$ is $9c{{m}^{2}}$and area of $\vartriangle XYZ$is $16c{{m}^{2}}$ and $QR=2.1cm$ As we know that, the ratio of areas of...

11. ΔXY $\vartriangle XYZ\sim \vartriangle DEF$ . If $YZ=3cm$ , $EF=4cm$ and area of $\vartriangle XYZ=54cm$ , find the area of ΔDEF.

Class 10, Frank, Guides, ICSE, Maths, Similarity

Given that, $\vartriangle XYZ\sim \vartriangle DEF$ and $YZ=3cm$, $EF=4cm$ and area of $\vartriangle XYZ=54c{{m}^{2}}$2 As we know that, the ratio of areas of two similar triangles is equal to the...

9. $\vartriangle XYZ\sim \vartriangle PQR$ such that $XY=1.8cm$ and $PQ=2.1cm$ . Find the ratio of areas of $\vartriangle XYZ$ and $\vartriangle PQR$ .

Class 10, Frank, ICSE, Maths, Similarity

given that, $\vartriangle XYZ\sim \vartriangle PQR$ and $XY=1.8cm$, $PQ=2.1cm$ ratio of areas of two similar triangles is equal to the ratio of square of their corresponding sides. Area of...

8. In ΔPQR, D and E are points on the sides PQ and QR respectively. If $PD=4cm$ , $DQ=4.5cm$ , $PE=6.4cm$ and $ER=7.2cm$ , find if DE is parallel to QR or not.

Class 10, Frank, ICSE, Maths, Similarity

From the question it is given that, $PD=4cm$, $DQ=4.5cm$, $PE=6.4cm$ and $ER=7.2 cm$ We have to show that, $DE||QR$ Consider the $\vartriangle PQR$, $PD/DQ=4/4.5$ $=40/45$ $PD/DQ=8/9$ … (i)...

7. D and E are points on the sides PQ and PR respectively of such that $PQ=5.6cm$ , $PD=1.4cm$ , $PR=7.2cm$ and $PE=1.8cm$ , show that DE is parallel to QR.

Class 10, Frank, ICSE, Maths, Similarity

From the question it is given that, $PQ=5.6cm$, $PD=1.4cm$, $PR=7.2cm$ and $PE=1.8cm$ We have to show that, $DE||QR$ Consider the $\vartriangle PQR$, $PD/DQ=1.4/4.2$ $=14/42$ $PD/DQ=1/3$ … (i)...

6. In $\vartriangle PQR$ , DE is drawn parallel to QR. If $PD:DQ=2:3$ , $DE=6cm$ and $PE=3.6cm$ find QR and QR

Class 10, Frank, ICSE, Maths, Similarity

From the question it is given that, $PD:DQ=2:3$, $DE=6cm$ and $PE=3.6cm$ Now consider the $\vartriangle PQR$, $DE||QR$ … [given] From Basic Proportionality Theorem (BPT) $PD/DQ=PE/ER$ $2/3=3.6/m$...

5. PS and QR are two straight lines intersecting at O. SR and QP are perpendiculars from Q and R on PS. If $PQ=6cm$ , $SR=9cm$ , $PS=20cm$ and $QR=25cm$ , find the lengths of PO, QO, RO and SO.

Class 10, Frank, ICSE, Maths, Similarity

question it is given that, $PQ=6cm$, $RS=9cm$, $PS=20cm$ and $QR=25cm$ We have to find the lengths of PO, QO, RO and SO. Consider $\vartriangle POQ$ and $\vartriangle ROS$, $\angle OPQ=\angle OSR$ …...

4. DEFG and ABCD are similar figures. $DE=12cm$ , $EF=xcm$ , $EG=15cm$ , $GD=10cm$ , $AB=8cm$ , $BC=5cm$ , $CD=mcm$ and $DA=ncm$ . Calculate the values of x, m and n.

Class 10, Frank, ICSE, Maths, Similarity

Given, quadrilateral DEFG $\sim $ quadrilateral ABCD. $DE=12cm$, $EF=xcm$, $FG=15cm$, $GD=10cm$, $AB=8cm$, $BC=5cm$, $CD=mcm$ and $DA=ncm$. $DE/AB=EF/BC=FG/DC=GD/DA$ $12/8=x/5=15/m=10/n$ Consider,...

In the adjoining figure, ABC is a triangle. DE is parallel to BC and AD/DB = 3/2,
(i) Determine the ratios AD/AB, DE/BC0
(ii) Prove that ∆DEF is similar to ∆CBF. Hence, find EF/FB.
(iii) What is the ratio of the areas of ∆DEF and ∆CBF?

In the given figure, CM and RN are respectively the medians of ∆ABC and ∆PQR. If ∆ABC ~ ∆PQR, prove that:
(i) ∆AMC ~ ∆PQR
(ii) CM/RN = AB/PQ
(iii) ∆CMB ~ ∆RNQ

State which pairs of triangles in the figure given below are similar. Write the similarity rule used and also write the pairs of similar triangles in symbolic form (all lengths of sides are in cm):

10. $\vartriangle XYZ\sim \vartriangle PQR$ , AD and PS are altitudes from X and P on sides YZ and QR respectively. If $XD:PS=4:9$ , find the ratio of the areas of $\vartriangle XYZ$ and $\vartriangle PQR$ .

15. In $\vartriangle XYZ$ , $DE||YZ$ ; DZ and EY intersect at F, if $DE/YZ=2/7$ , find area of $\vartriangle FDE$ /area of $\vartriangle FYZ$

14. in $\vartriangle XYZ$ , $MN||YZ$ (a) If $XN:YZ=5:8$ , find the ratio of area of $\vartriangle (b) If$ XY/XM=9/4 $, find area of trapezium MBCN/area of$ \vartriangle XYZ$.

13. Prove that the area of $\vartriangle XYE$ described on one side XY of a square WXYZ is one half the area of the similar $\vartriangle WYF$ described on the diagonal WY.

12. . If area of $\vartriangle PQR$ is $9c{{m}^{2}}$ and area of $\vartriangle XYZ$ is $16c{{m}^{2}}$ and if $QR=2.1cm$ , find the length of YZ.

11. ΔXY $\vartriangle XYZ\sim \vartriangle DEF$ . If $YZ=3cm$ , $EF=4cm$ and area of $\vartriangle XYZ=54cm$ , find the area of ΔDEF.

9. $\vartriangle XYZ\sim \vartriangle PQR$ such that $XY=1.8cm$ and $PQ=2.1cm$ . Find the ratio of areas of $\vartriangle XYZ$ and $\vartriangle PQR$ .

8. In ΔPQR, D and E are points on the sides PQ and QR respectively. If $PD=4cm$ , $DQ=4.5cm$ , $PE=6.4cm$ and $ER=7.2cm$ , find if DE is parallel to QR or not.

7. D and E are points on the sides PQ and PR respectively of such that $PQ=5.6cm$ , $PD=1.4cm$ , $PR=7.2cm$ and $PE=1.8cm$ , show that DE is parallel to QR.

6. In $\vartriangle PQR$ , DE is drawn parallel to QR. If $PD:DQ=2:3$ , $DE=6cm$ and $PE=3.6cm$ find QR and QR

5. PS and QR are two straight lines intersecting at O. SR and QP are perpendiculars from Q and R on PS. If $PQ=6cm$ , $SR=9cm$ , $PS=20cm$ and $QR=25cm$ , find the lengths of PO, QO, RO and SO.

4. DEFG and ABCD are similar figures. $DE=12cm$ , $EF=xcm$ , $EG=15cm$ , $GD=10cm$ , $AB=8cm$ , $BC=5cm$ , $CD=mcm$ and $DA=ncm$ . Calculate the values of x, m and n.

3. ΔABC is similar to ΔPQR. If $AB=6cm$ , $BC=9cm$ , $PQ=9cm$ and $PR=10.5cm$ , find the lengths of AC and QR.

2. In $\vartriangle PQR$ , MN is drawn parallel to QR. If $PM=x$ , $MQ=(x-2)$ and $NR=(x-1)$ , find the value of x.

1. In $\vartriangle ABC$ , DE is parallel to BC and $AD/DB=2/7$ . If $AC=5.6$ , find AE.

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