Theorem 2: Triangles

Q.

Let $OA=a,OB=10a+2bandOC=b$ where $O,AandC$ are non-collinear points. Let $p$denote the area of the quadrilateral $OABC$ and $q$ denote the area of the parallelogram with $OA$ and $OC$ as adjacent sides. If $p=kq$ then $k$ is equal to?

1. $4$

2. $6$

3. $0$

4. $1$

Q.

E is the mid-point of a median AD of $\Delta ABC$ and BE is produced to meet AC at F. Show that $AF=\frac{1}{3}AC$.

Q. Question 4
ABCD is a trapezium in which AB || DC, BD is a diagonal and E is the mid - point of AD. A line is drawn through E parallel to AB intersecting BC at F ( See the given figure). Show that F is the mid - point of BC.

Q.

Show that a diagonal divides a parallelogram into two triangles of equal area.

Q.

Each of the median of a triangle is divided by the centroid in the ratio ___ from the side of vertices.

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.

A point O is taken inside an equilateral $\Delta ABC$. If $OM\perp AC$, $OL\perp BC$ and $ON\perp AB$ such that OL = 14 cm, OM = 10 cm and ON = 6 cm, find the area of $\Delta ABC$.

1. $200\sqrt{3}c{m}^2$

2. $300\sqrt{3}c{m}^2$

3. $250\sqrt{3}c{m}^2$

4. $100\sqrt{3}c{m}^2$

Q.

In the given figure, PQRS and PXYZ are two parallelograms of equal area. which of the following statements is true?

1. SX || YR
2. area $\Delta YSX$ =area $\Delta SXR$
3. All of the above.
4. area$\Delta YSR$ =area $\Delta YXR$

Q.

Diagonals AC and BD of a trapezium ABCD with AB $\parallel$ DC intersect each other at O.then, ar ($△$AOD) = ___.

1. Area ($△$BOC)

2. Area ($△$AOD)

3. Area ($△$BOD)

4. None of the above

Q. A triangle and a parallelogram are on the same base and between the same parallel. The area of triangle is $54c{m}^2$. Find the area of parallelogram.

1. $54c{m}^2$
2. $108c{m}^2$
3. $27c{m}^2$
4. $28c{m}^2$

Q. If A, B, C, D are mid-points of sides of parallelogram PQRS, area of PQRS is always ________ the area of shape formed by joining A, B, C, D.

1. thrice
2. half
3. one-third
4. twice

Q.

ABCD is a parallelogram whose area is 60 $c{m}^2$. The area of triangle AEB is __ $c{m}^2$.

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Q.

AE || BC and D is the mid-point of BC. If area $△ABC$ = 84 $c{m}^2$, what is area of $△BDE$?

1. 84 $c{m}^2$
2. 42$c{m}^2$
3. 63 $c{m}^2$
4. 21$c{m}^2$

Q.

In the following figure, ABCD is a parallelogram and BC is produced to a point Q such that AD = CQ. If AQ intersect DC at P, then area of($△$BPC) equal to

1. area of(ΔBPC) = (1/3) area of (ΔDPQ)

2. area of(ΔBPC) = (3/8) area of (ΔDPQ)

3. area of (ΔBPC) = (1/4) area of (ΔDPQ)

4. area of (ΔBPC) = area of (ΔDPQ)

Q.

In the figure shown, the area of triangle APB is

1. 6 $c{m}^2$

2. 12 $c{m}^2$

3. 24 $c{m}^2$

4. 48 $c{m}^2$

Q. Question 17
Prove that the line joining the mid-point of the diagonals of a trapezium is parallel to the parallel sides of the trapezium.

Q.

In the given figure, F and E are points on the side AD of $\Delta ABD$. Through F a line is drawn parallel to AB to meet BD at the point C. Area of quadrilateral BCEF is equal to ________ .

1. area of $\Delta ACE$

2. area of $\Delta BFC$

3. area of $\Delta ABC$

4. area of $\Delta ABD$

Q.

If D and E are points on sides AB and AC respectively of $△$ABC such that ar ($△$DBC) = ar ($△$EBC), then DE $\parallel$ BC.

1. False

2. True

Q.

ABCD is a parallelogram and P, Q are the midpoints of DC and AB respectively. Then, area of parallelogram AQPD is equal to area of triangle ADB

1. True

2. False

Q.

If the area of parallelogram PQRS is 32 cm² and line XY || AS , then the area of triangle ACS is____

1. Data Insufficient

2. 32

3. 8

4. 16

Q.

Two triangles having the same base (or equal bases) and equal areas need not necessarily lie between the same parallels.

1. True

2. False

Q.

In the given figure AD is median of $\Delta ABC$, DE is median of $\Delta ABD$ and EF is median of $\Delta BED$. If area of $\Delta ABC$ is $128c{m}^2$, then area of $\Delta BEF$ is

1. $8c{m}^2$

2. $16c{m}^2$

3. $32c{m}^2$

4. $64c{m}^2$

Q. Let ABC be a triangle of area 24 sq. units and PQR be the triangle formed by the mid-points of the sides of Δ ABC. Then the area of ΔPQR is

(a) 12 sq. units

(b) 6 sq. units

(c) 4 sq. units

(d) 3 sq. units

Q. Diagonal AC and BD of trapezium ABCD, in which AB || DC, intersect each other at O. The triangle which is equal in area of ΔAOD is

(a) ΔAOB

(b) ΔBOC

(c) ΔDOC

(d) ΔADC

Q. Two triangles have the same base and between same parallels. Area of one triangle is 36 sqcm. Area of 2nd is .

1. 18 sq.cm
2. 9 sq.cm
3. 36 sq.cm

Q.

ABCD is a rectangle of area $40c{m}^2$. If X is any point on AB, then what is the area of $\Delta XCD$ ?

1. 
2. 
3. 
4. 

Q.

In the rectangle ABCD, O is any point inside the rectangle. If area( ΔAOD) = 30 $c{m}^2$ and area( ΔBOC) = 60 $c{m}^2$, area of the rectangle ABCD is ___ $c{m}^2$.

Q.

The given figure shows parallelogram ABCD. Points P and Q trisect the side AB, then:

1. Area of ΔDPQ = (1/3) area of (ABCD)

2. Area of ΔDPQ = (2/5) area of (ABCD)

3. Area of ΔDPQ = (1/6) area of (ABCD)

4. Area of ΔDPQ = (4/5) area of (ABCD)

Q. Trapezium PQRS and $△$MNS have same base and lying in between same parallels.

1. True
2. False

Q.

In the given figure, if BD=DC, AP=PC and area $\Delta BPC=20c{m}^2,$ then area $\Delta ABD=$ ___$c{m}^2$

1. 20

2. 40

3. 50

4. 10