Basic Proportionality Theorem

Q. ABCD is a quadrilateral in which the bisectors of angle-A and angle-C meet DC produced at Y and BA produced at X, respectively. Prove that angle-X+angle-Y=1/2(angle-A+angle-C)

Q. in square ABCD AC and BD meet at O the angle bisector of \angle CAB meets BD at F, meets BC at G .Find the ratio of OF toCG

Q.

If PQ $\parallel$ BC and PR $\parallel$CD in the given figure, then prove that $\frac{AR}{AD}$= $\frac{AQ}{AB}$.

Q.

In the given figure $AB\parallel EF\parallel DC$; AB = 67.5 cm. DC = 40.5 cm and AE = 52.5 cm. Find the length of EC.

1. EC = 24.4 cm

2. EC = 70.6 cm

3. EC = 45.7 cm

4. EC = 31.5 cm

Q.

In below shown figure, $\frac{PS}{SQ}$= $\frac{PT}{TR}$ and$\angle$PST= $\angle$PRQ. Then $\Delta$PQR is an isosceles Triangle.

1. False

2. True

Q.

The perimeters of two similar triangles are $30cm$ and $20cm$ respectively. If one side of the first triangle is $12cm$ , determine the corresponding side of the second triangle.

Q.

In $△ABC,AD,BE$ and $CF$ are medians, and $G$ is the centroid. If $AD=15cm$, find $AG$.

1. 10

Q. In the given figure, AD = DB and AE = EC. The ratio, BC : DE = ___.

1. 8:1
2. 4:1
3. 1:1
4. 2:1

Q. Question 8 (i)
In right triangle ABC, right angled at C, M is the mid-point of hypotenuse AB. C is joined to M and produced to a point D such that DM = CM. Point D is joined to point B (see the given figure). Show that:
$\Delta AMC\cong \Delta BMD$

Q. In $\Delta ABC$, if D is a point on the side AB such that $\frac{AB}{AD}=3$. Find $\frac{AD}{DB}$.

1. $3$
2. $\frac{1}{3}$
3. $2$
4. $\frac{1}{2}$

Q. Question 15
Diagonals AC and BD of a quadrilateral ABCD intersect at O in such a way that ar(AOD) = ar(BOC). Prove that ABCD is a trapezium.

Q.

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

Q.

Match the following table:

$\begin{array}{cc}\text{Theorem Statement}& \text{Theorem Name}\\ \text{(a) The line segment joining the mid-points of}& \text{(i) Basics proportionality theorem}\\ \text{two sides of a triangle is parallel to the third side}& \\ \text{(b) A line parallel to one side of a triangle divides}& \text{(ii) Mid point theorem}\\ \text{the other two sides into parts of equal}& \\ \text{poportion}& \\ \text{(c) If a line divides the any tw o sides of a triangle}& \text{(iii) Converse of Basic porpotionality theorem}\\ \text{in the same ratio, then the line must be parallel}& \\ \text{to the third side.}& \end{array}$

1. (i) - b; (ii) - a; (iii) - c; (iv) - d

2. (i) - a; (ii) - b; (iii) - c; (iv) - d

3. (i) - b; (ii) - c; (iii) - a; (iv) - d

4. (i) - d; (ii) - a; (iii) - c; (iv) - b

Q. In $\Delta$ ABC, AC = 15 cm and DE ll BC. If $\frac{AB}{AD}=3$, find EC.

1. 2.5 cm
2. 10 cm
3. 9 cm
4. 5 cm

Q. In $\Delta$ ABC, DE $\parallel$ BC. Then according to Basic Proportionality Theorem

1. $\frac{AD}{DB}=\frac{AE}{EC}$
2. $\frac{AB}{DB}=\frac{AE}{EC}$
3. $\frac{AD}{DB}=\frac{AC}{EC}$
4. $\frac{AB}{DB}=\frac{EC}{AC}$

Q.

In the figure, PQRS is a parallelogram with and QR = 10 cm. L is a point on PR. RL : LP = 2 : 3. QL produced meets RS and PS produced at M. Select the statements that are true.

1. $PN=50\text{cm}$

2. $RM=20\frac{2}{3}\text{cm}$.

3. $RM=10\frac{2}{3}\text{cm}$.

4. $PN=15\text{cm}$

Q. In figure DE$\parallel$BC, AD=2.4cm, AE=3.2cm, CE=4.8cm The value of BD is

1. None of these
2. 4.2 cm
3. 4.0 cm
4. 3.6 cm

Q. In $\Delta$ ABC, DE $\parallel$ BC. Then which of the following is stated by Basic Proportionality Theorem?

1. $\frac{AD}{DB}=\frac{AE}{EC}$
2. $\frac{AB}{DB}=\frac{AE}{EC}$
3. $\frac{AD}{DB}=\frac{AC}{EC}$
4. $\frac{AB}{DB}=\frac{EC}{AC}$

Q. 15. ABCD is a parallelogram and APQ is a straight line meeting BC at P and DC produced atQ. prove that the rectangle obtained by BP and DQ is equal to the rectangle contained by AB and BC.

Q. In a parallelogram ABCD, E and F are the mid-points of sides AB and CD respectively (see the given figure).

Which of the following options is correct?

1. All of the above
2. BQ = PD
3. BQ = PQ
4. PQ = PD

Q.

In $\Delta$ ABC, DE $\parallel$ BC. Then which of the following is stated by Basic Proportionality Theorem?

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

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

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

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

Q.

In triangle ABC, D is a point on AB and E is a point on AC such that DE || BC. If $\frac{AD}{AB}$ = $\frac{AE}{x}$, Then $x$ is

___.

Q. In the given figure alongside , $BA\parallel DF,CA\parallel EG\&BD\parallel EC$ .
prove that
i) $BG=DF$
ii) $EG=CF$

Q. In the figure given $\Delta ABC$ is a right isosceles triangle with right angle at C CD is a parallel to AB and BD = BA. The degree measure of $\angle DBC$ equals:

1. ${10}^o$
2. ${15}^o$
3. ${20}^o$
4. ${25}^o$

Q.

Diagonals AC and BD of a trapezium ABCD with AB $\parallel$ DC intersect each other at O.
Prove that ar ($△$AOD) = ar ($△$BOC). [1 MARK]

Q. In a triangle $△ABC$, points P, Q and R are the mid-points of the sides AB, BC and CA respectively. If the area of the triangle ABC is 20 sq.units, then area of the triangle PQR equal to:

1. $5\sqrt{3}$ sq. units
2. $10$ sq. units
3. $5$ sq. units
4. $5.5$ sq. units

Q. ABC is a triangle. If DE || BC and $DE=2\times BC$, then burst the correct equalities.

1. AD = DB
2. AD = AB
3. $AD=2\times AB$
4. AE = AC
5. AE = EC
6. $AE=2\times AC$
7. ∠ADE = ∠ABC
8. ∠AED+∠ACB
   = 180°

Q. In parallelogram $ABCD$, the bisector of angle $A$ meets $CB$ in $P$ and the bisector of angle $B$ meets $DA$ in $Q$ and $AB=2AD$.Then $\angle ABP=xCBP$. Find the value of $x$.

Q. The given figure shows an equilateral triangle $ABC$ with each side $15$ cm. Also $DE\parallel BC$, $DF\parallel AC$, and $EG\parallel AB$. If $DE+DF+EG=20$ cm, then the length of $FG$ is

1. $5$ cm
2. $10$ cm
3. $2.5$ cm
4. $7.5$ cm

Q. In the given figure of $△ABC$, let $\angle A={50}^o,B={60}^o,$ and $C={70}^o$. If mid-point of side $AB,BC$ and $CA$ are $D,E$ and $F$ respectively then find $\angle DEF$