Converse of Basic Proportionality Theorem

Q.

$\mathrm{ABCD}$ is a trapezium in which $\mathrm{AB}$ parallel $\mathrm{DC}$and its diagonals intersect each other at point $\mathrm{O}$ . show that $\frac{\mathrm{AO}}{\mathrm{BO}}=\frac{\mathrm{CO}}{\mathrm{DO}}$

Q.

In the adjoining figure. ABC is a triangle in which AB = AC. If D and E are points on AB and AC respectively such that AD = AE, show that the points B, C, E and D are concyclic.

Q.

In figure, $△FEC\cong △GBD$ and $\angle 1=\angle 2$. Prove that $△ADE\sim △ABC$.

Q.

What is the difference between lemma and axiom?

Q. In ∆ABC, D and E are points on side AB and AC respectively such that DE || BC and AD : DB = 3 : 1. If EA = 3.3 cm, then AC =

(a) 1.1 cm
(b) 4 cm
(c) 4.4 cm
(d) 5.5 cm

Q.

In the given figure, If $PQ\parallel RS$, prove that $\Delta POQ\sim \Delta SOR$.
[2 MARKS]

Q. In an equilateral triangle ABC, AD is parpendicular to BC . Prove that 3AB² = 4AD².

Q.

Let ABC be a triangle and D and E be two points on side AB such that AD = BE. If DP $\parallel$ BC and EQ $\parallel$ AC, then prove that PQ $\parallel$ AB.

Q.

In triangle DEF, GH is a line parallel to EF cutting DE in G and DF in H. If DE=16.5, DH=5, HF=6 GE=?

1. 9

2. 8

3. 10

4. 7.5

Q. If ABC is a right triangle right-angled at B and M, N are the mid-points of AB and BC respectively, then 4(AN² + CM²) =

(a) 4 AC²
(b) 5 AC²
(c) $\frac{5}{4}{\mathrm{AC}}^2$
(d) 6 AC²

Q.

In the following figure, if LM || CB and LN || CD, prove that

Q.

P and Q are points on the sides AB and AC respectively of a $\Delta ABC$. If AP = 2 cm, PB = 4 cm, AQ = 3 cm, and QC = 6 cm, show that BC = 3PQ.

Q.

D and E are respectively the points on the sides AB and AC of a triangle ABC , such that AE =5cm AC=7.5 cm and DE =4.2cm and DE//BC , find the length of BC.

Q.

The ratio of corresponding sides of two similar triangles $△ABC$ ~ $△PQR$ is given to be 4:7. Find the side QR if BC is given to be 3.5cm.

Q.

State the converse of Thales' theorem.

Q.

Mid point theorem is a special case of converse of basic proportionality theorem. (T/F)

1. True

2. False

Q.

In figure, if AB || DC and AC, PQ intersect each other at the point O. Prove that OA.CQ = OC.AP.

Q. In Fig. 7, AB $\perp$ BC, FG $\perp$ BC and DE $\perp$ AC. Prove that Δ ADE $~$ ΔGCF

Q. In ∆ABC X is middle point of AC.if XYparallel to AB then prove that Y is middle point of AB

Q. Question 5
S and T are point on sides PR and QR of $\Delta PQR$ such that $\angle P=\angle RTS$. Show that $\Delta RPQ\sim \Delta RTS$.

Q.

In Fig. 7.137, AB || QR. Find the length of PB.

Q.

In $\Delta ABC$, point D and E lies on the line AB and AC respectively as shown in the figure. Find the measure of $\angle AED$.

1. ${65}^{\circ }$

2. ${40}^{\circ }$

3. ${75}^{\circ }$

4. ${70}^{\circ }$

Q.

Question 7 (iii)
In the below figure, altitudes AD and CE of $\Delta ABC$ intersect each other at the point P. Show that


(iii) $\Delta AEP\sim \Delta ADB$

Q. Question 3
A and B are respectively the points on the sides PQ and PR of a $\Delta$ PQR such that PQ = 12.5 cm, PA = 5 cm, BR = 6 cm and PB = 4 cm. Is AB||QR? Give reasons for your answer.

Q. Question 9 (ii)
In the following figure, ABC and AMP are two right triangles, right angled at B and M respectively, prove that:

$\left(ii\right)\frac{CA}{PA}=\frac{BC}{MP}$

Q. Question 6
In the below figure, if $\Delta ABE\cong \Delta ACD$, show that $\Delta ADE\sim \Delta ABC$.

Q.

Question 1 (iv)
State whether the pair of triangles in the below figure is similar. Write the similarity criterion used by you for answering the question and also write the pair of similar triangles in the symbolic form:

Q. Question 11
In figure, if $\angle D=\angle C$, then is it true that $\Delta ADE\sim \Delta ACB$, ? Why?

Q. Construct $△$ PYQ such that, PY $=6.3cm,YQ=7.2cm,PQ=$ $5.8cm.$ If $=YZYQ=65$ then construct $△XYZ$ similar to $△PYQ$

Q. Question 9 (i)
In the following figure, ABC and AMP are two right triangles, right angled at B and M respectively, prove that

$\left(i\right)\Delta ABC\sim \Delta AMP$