Basic Proportionality Theorem and Its Forms

Q. $\Delta ABC\sim \Delta PQR$, ∠B = 50° and ∠C = 70° . Then ∠P is equal to:

1. ${40}^{\circ }$
2. ${70}^{\circ }$
3. ${50}^{\circ }$
4. ${60}^{\circ }$

Q. Using the information in the given figures, match the value of x in each case.

1. 3.2 cm
2. 18 cm
3. 12 cm

Q. Select the appropriate alternative.
(1) In ∆ABC and ∆PQR, in a one to one correspondence $\frac{\mathrm{AB}}{\mathrm{QR}}=\frac{\mathrm{BC}}{\mathrm{PR}}=\frac{\mathrm{CA}}{\mathrm{PQ}}$ then
(2) If in ∆DEF and ∆PQR, ∠D ≅ ∠Q, ∠R ≅ ∠E then which of the following statements is false?

(A) $\frac{\mathrm{EF}}{\mathrm{PR}}=\frac{\mathrm{DF}}{\mathrm{PQ}}$ (B) $\frac{\mathrm{DE}}{\mathrm{PQ}}=\frac{\mathrm{EF}}{\mathrm{RP}}$

(C) $\frac{\mathrm{DE}}{\mathrm{QR}}=\frac{\mathrm{DF}}{\mathrm{PQ}}$ (D) $\frac{\mathrm{EF}}{\mathrm{RP}}=\frac{\mathrm{DE}}{\mathrm{QR}}$



(3) In ∆ABC and ∆DEF ∠B = ∠E, ∠F = ∠C and AB = 3DE then which of the statements regarding the two triangles is true ?
(A)The triangles are not congruent and not similar
(B) The triangles are similar but not congruent.
(C) The triangles are congruent and similar.
(D) None of the statements above is true.


(4) ∆ABC and ∆DEF are equilateral triangles, A (∆ABC) : A (∆DEF) = 1 : 2
If AB = 4 then what is length of DE?
(A) $2\sqrt{2}$
(B) 4
(C) 8
(D) $4\sqrt{2}$



(5) In the given figure, seg XY || seg BC, then which of the following statements is true?

(A) $\frac{\mathrm{AB}}{\mathrm{AC}}=\frac{\mathrm{AX}}{\mathrm{AY}}$ (B) $\frac{\mathrm{AX}}{\mathrm{XB}}=\frac{\mathrm{AY}}{\mathrm{AC}}$

(C) $\frac{\mathrm{AX}}{\mathrm{YC}}=\frac{\mathrm{AY}}{\mathrm{XB}}$ (D) $\frac{\mathrm{AB}}{\mathrm{YC}}=\frac{\mathrm{AC}}{\mathrm{XB}}$

Q. In the following figure, if LM || CB and LN || CD, prove that $\frac{AM}{AB}=\frac{AN}{AD}$.

Q. AB is parallel to CD, EF intersects them at M and N. The bisectors of $\angle BMNand\angle MND$ meet at Q. If $\angle AME={80}^{\circ }$, then $\angle MQN$ is:

1. ${90}^{\circ }$
2. ${70}^{\circ }$
3. ${80}^{\circ }$
4. ${60}^{\circ }$

Q. In a triangle ABC, a line is drawn parallel to BC such that it intersects the other two sides at the points P, Q. Which of the following is true regarding the given information?

1. A triangle ABC is similar to BPQ.
2. A triangle ABC is similar to CPQ.
3. A triangle ABC is similar to APQ.
4. Triangles ABC andAPQ need not be similar.

Q.

In the following figure, AB, CD and EF are perpendicular to the straight line BDF.

If AB = x units and CD = z units and EF = y units, then which of the following is/are true?

1. $\frac{1}{x}+\frac{1}{y}=\frac{1}{z}$

2. $\frac{1}{x}+\frac{1}{z}=\frac{1}{y}$

3. $\frac{1}{x}-\frac{1}{z}=\frac{1}{y}$

4. $-\frac{1}{x}-\frac{1}{z}=\frac{1}{y}$

Q.

BO and CO are respectively the bisectors of angle B and C of ∆ABC. AO produced meets BC at P. Show that AO/BP = AO/OP , AC/CP = AO/OP ,

AB/AC = BP/PC , AP is the bisector of angle BAC.

Q.

D is a point on the side BC of a triangle ABC such that ∠ADC = ∠BAC. Show that

Q. 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.

Q. In the following figure, DE||AC and DF||AE. Prove that $\frac{BF}{FE}=\frac{BE}{EC}$ .

Q. In the figure given below, point D divides AB in the ratio 3 : 5 and DE is parallel to BC. If $AE=4.8$ cm, then find the length of AC.

1. 4.8
2. 12
3. 12.8
4. 8.8

Q.

Prove that two isosceles triangles having an angle of one equal to the corresponding angle of the other are similar.

Q. In the figure given below, point D divides AB in the ratio 3 : 5 and DE is parallel to BC. If AE = 4.8 cm, then find the length of AC.

1. 12 cm
2. 4.8 cm
3. 12.8 cm
4. 8.8 cm

Q. If in ∆ABC and ∆PQR, we have: $\frac{AB}{QR}=\frac{BC}{PR}=\frac{CA}{PQ}$, then
(a) ∆PQR ∼ ∆CAB
(b) ∆PQR ∼ ∆ABC
(c) ∆CBA ∼ ∆PQR
(d) ∆BCA ∼ ∆PQR

Q.

In fig, $\frac{PS}{SQ}$ = $\frac{PT}{TR}$ and $\angle$PQR. Prove that PQR is an isosceles triangle.

Q. If triangles ABC and ADE are similar then AB : AD =

1. AC : CE
2. BD : DA
3. BC : AE
4. BC : DE

Q. The sides of a triangle are $10\frac{1}{2}$ , $16\frac{1}{2}$ , and 9. An angle bisector meets the side of length 9. Find the lengths 𝒙 and 𝒚.

1. 𝒙 = 5.5 and 𝒚 = 3.5
2. 𝒙 = 3.5 and 𝒚 = 3.5
3. 𝒙 = 6.5 and 𝒚 = 5.5
4. None of the above

Q.

In the above figure, AB=c, BC =a, AC=b , AD=y, DB=p. Check which of the following options is correct?

1. cy=ap

2. ay=cp

3. ac=by

4. cy=ab

Q. In $\Delta ABC$, AC = 15 cm and DE || BC. If $\frac{AD}{DB}=\frac{1}{2}$, then AE=___

1. 5 cm
2. 7.5 cm
3. 10 cm
4. 12.5 cm

Q. If in two triangles ABC and DEF, $\frac{\mathrm{AB}}{\mathrm{DE}}=\frac{\mathrm{BC}}{\mathrm{FE}}=\frac{\mathrm{CA}}{\mathrm{FD}}$, then

(a) ∆FDE ∼ ∆CAB
(b) ∆FDE ∼ ∆ABC
(c) ∆CBA ∼ ∆FDE
(d) ∆BCA ∼ ∆FDE

Q.

In the above figure, $AB\parallel MN\parallel PC$, then which of the following will be true?

1. $\frac{1}{MN}+\frac{1}{PC}=\frac{1}{AB}$

2. $\frac{1}{AB}+\frac{1}{MN}=\frac{1}{PC}$

3. $\frac{1}{AB}+\frac{1}{PC}=\frac{1}{MN}$

4. $\frac{1}{AB}-\frac{1}{PC}=\frac{1}{MN}$

Q. In the figure given below, point D divides AB in the ratio 3 : 5 and DE is parallel to BC. If AE = 4.8 cm, then find the length of AC.

Q.

In the figure shown below, $\frac{PS}{SQ}=\frac{PT}{TR}$ and $\angle PST=\angle PRQ.$ Then $△PQR$ is __ triangle.

Q.

In the figure, $D$ and $E$ trisect $BC$, Prove that $8{AE}^2=3{AC}^2+5{AD}^2$
[4 Marks]

Q.

Given two triangles ABC and DEF, AG and DH are perpendiculars on BC and EF respectively. ∠B=∠E, B=5, DE=8. What is AG/DH = ?

1. 

2. 

3. 

4. 

Q. In the figure, PQ || AB, CQ = 4.8 cm, QB = 3.6 cm and AB = 6.3 cm. If AP = x, length of AC in terms of x is .

1. $\frac{8}{3}x$
2. $\frac{5}{3}x$
3. $\frac{4}{3}x$
4. $\frac{7}{3}x$

Q. In the given figure (i) and. (ii) DE $\parallel$ BC. Find EC in(i) and AD in (ii).

1. $EC=24cm$ , $AD=2cm$
2. $EC=2cm$ , $AD=2.4cm$
3. $EC=2.4cm$ , $AD=2cm$
4. $EC=2cm$ , $AD=24cm$

Q. In the following figure, if LM || CB and LN || CD, prove that $\frac{AM}{AB}=\frac{AN}{AD}$.

Q. In the following figure, DE||OQ and DF||OR, show that EF||QR.