Basic Proportionality Theorem

Q. If D, E and F are respectively the midpoints of sides AB, BC and CA of △ABC then what is the ratio of the areas of △DEF and △ABC?

Q. In the given figure, $AB||DEandBD||EF.Then,D{C}^2=?$

1. 
   $CF\times AC$
2. 
   $CF÷AC$
3. 
   $DF\times EB$
4. 
   $DF÷EB$

Q.

Question 9
ABCD is a trapezium in which AB || DC and its diagonals intersect each other at the point O. Show that $\frac{AO}{BO}=\frac{CO}{DO}$.

Q. SAT is a triangle. P and Q are points on SA and ST respectively such that PQ || AT. If M is a point on SA such that MQ || PT, then which of the following is always true?

1. $S{Q}^2=SA\times PM$
2. $S{P}^2=ST\times SM$
3. $S{P}^2=SA\times PM$
4. $S{P}^2=SA\times SM$

Q.

The diagonals of a quadrilateral $ABCD$ intersect each other at the point $O$ such that $\frac{AO}{BO}=\frac{CO}{DO}$ . Show that $ABCD$ is a trapezium.

Q.

$P$ is a point on side $BC$ of a parallelogram $ABCD$. If $DP$ produced meets $AB$ produced at point $L$, prove that :
(i) $DP:PL=DC:BL$.
(ii) $DL:DP=AL:DC$.

Q.

All the values of $m$ for which both roots of the equation $x^2-2mx+m^2-1=0$ are $>-2$ but $<4$ lie in the interval

1. $m>3$

2. $–1<m<3$

3. $1<m<4$

4. $–2<m<0$

Q. State basic proportionality theorem and its converse.

Q. Question 7
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.

Q.

The logical statement $\left(\mathrm{p}\Rightarrow \mathrm{q}\right)\wedge \left(\mathrm{q}\Rightarrow ~\mathrm{p}\right)$ is equivalent to

1. $~\mathrm{p}$

2. $\mathrm{p}$

3. $\mathrm{q}$

4. $~\mathrm{q}$

Q. What is thales theorem?

Q.

$AE$ is the bisector of the exterior $\angle CAD$ meeting $BC$ produced in $E$. If $AB=10cm$, $AC=6cm$ and $BC=12cm$, find $CE$.

Q. Question 14
In figure , PA, QB, RC and SD are all perpendiculars to a line l, AB = 6cm, BC = 9cm, CD = 12cm and SP = 36cm. Find PQ, QR and RS.

Q.

A $10$ cm long rod $AB$ moves with its ends on two mutually perpendicular straight lines $OX$ and $OY$. If the end $A$ be moving at the rate of $2$ cm/sec, then when the distance of $A$ from $O$ is $8$cm, the rate at which the end $B$ is moving, is

1. $\frac{8}{3}$cm / sec

2. $\frac{4}{3}$cm / sec

3. $\frac{2}{9}$cm / sec

4. None of these

Q. Question 4
In the following figure, DE||AC and DF||AE. Prove that $\frac{BF}{FE}=\frac{BE}{EC}$ .

Q. D is the mid-point of side BC of a ∆ABC. AD is bisected at the point E and BE produced cuts AC at the point X. Prove that BE = EX = 3 : 1

Q.

In triangle ABC, if DE is parallel to BC, then which of the following is stated by Basic Proportionality Theorem?

1. $\frac{AB}{DB}=\frac{AE}{EC}$

2. $\frac{AD}{DB}=\frac{AC}{EC}$

3. $\frac{AD}{DB}=\frac{AE}{EC}$

4. $\frac{AB}{DB}=\frac{EC}{AC}$

Q.

M and N are points on the sides PQ and PR respectively of a $\Delta PQR$. For each of the following cases, whether MN || QR:
(i) PM = 4 cm, QM = 4.5 cm, PN = 4 cm, NR = 4.5 cm
(ii) PQ =1.28 cm, PR = 2.56 cm, PM = 0.16 cm, PN = 0.32 cm

Q.

In the given figure, side BC of $\Delta ABC$ is bisected at D and O is any point on AD, BO and CO produced meet AC and AB at E and F respectively, and AD is produced to X so that D is the midpoint of OX. Prove that AO : AX = AF : AB and show that EF || BC.

Q. In the figure given below DE || BC. If AD = 2.4 cm, DB = 3.6 cm, AC = 5 cm. Find AE.

Q.

In Fig. 7.141 DE || BC such that $AE=\left(\frac{1}{4}\right)AC$. If AB = 6 cm, find AD.

Q.

The following figure $GUNS$ is a parallelogram. Find $x$ and$y$. (Lengths are in $cm$)

Q.

$ABCD$ is a rhombus and $P,Q,R$ and $S$ are mid-points of the sides $AB,BC,CD$ and $DA$ respectively. Show that the quadrilateral $PQRS$ is a rectangle.

Q.

If LM $\parallel$ AB, AL=x-3, AC=2x, BM=x-2, BC=2x+3. What is value of AC?

__

Q.

D and E are points on the sides AB and AC respectively of a $\Delta ABC$ such that DE || BC
Find the value of x, when

(i) AD = x cm, DB = (x - 2) cm,

AE = (x + 2) cm and EC = (x - 1) cm.

(ii) AD = 4 cm, DB = (x - 4) cm, AE = 8 cm and EC = (3x - 19) cm.

(iii) AD = (7x - 4) cm, AE = (5x - 2) cm, DB = (3x + 4 ) cm and EC = 3x cm.

Q.

Question 5
In the following figure, DE||OQ and DF||OR, show that EF||QR.

Q.

The minimum value of $4e^{2x}+9e^{-2x}$ is

1. $11$

2. $12$

3. $10$

4. $14$

Q. Question 15
O is the point of intersection of the diagonals AC and BD of a trapezium ABCD with ABIIDC. Through O, a line segment PQ is drawn parallel to AB meeting AD in P and BC in Q. Prove that PO = QO.

Q.

Question:

In △ABC, AB = 3 and, AC = 4 cm and AD is the bisector of ∠A. Then, BD : DC is —

Solution:

BD/DC = AB/AC =34

So, BD : DC = 3 : 4.

In did not understand how we proved the similarity and how we are taking the sides proportionality. Plz explain. Thank you.

Q.

$\Delta ABC$ and $\Delta DBC$ lie on the same side of BC, as shown in the figure. From a point P on BC, PQ || AB and PR || BD are drawn, meeting AC at Q and CD at R respectively. Prove that QR || AD.