# Basic Proportionality Theorem

## Trending Questions

**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||DE and BD||EF.Then, DC2=?

CF×AC

CF÷AC

DF×EB

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 AOBO=CODO.

**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?

- SQ2=SA×PM
- SP2=ST×SM
- SP2=SA×PM
- SP2=SA×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

$m>3$

$\u20131<m<3$

$1<m<4$

$\u20132<m<0$

**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

$~\mathrm{p}$

$\mathrm{p}$

$\mathrm{q}$

$~\mathrm{q}$

**Q.**What is thales theorem?

**Q.**

AE is the bisector of the exterior ∠CAD meeting BC produced in E. If AB=10 cm, AC=6 cm and BC=12 cm, 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

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

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

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

None of these

**Q.**Question 4

In the following figure, DE||AC and DF||AE. Prove that BFFE=BEEC .

**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?

ABDB=AEEC

ADDB=ACEC

ADDB=AEEC

ABDB=ECAC

**Q.**

M and N are points on the sides PQ and PR respectively of a ΔPQR. 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 ΔABC 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=(14)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 ∥ 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 ΔABC 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 $4{e}^{2x}+9{e}^{-2x}$ is

$11$

$12$

$10$

$14$

**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.**

ΔABC and ΔDBC 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.