Constructions

Question

ΔABC and ΔABD are two triangles on the same base AB. If line segment CD is bisected by AB at O, show that ar (ΔABC) = ar (ΔABD).

$ar left parenthesis quad. space ABCD right parenthesis equals 1 half BD. left parenthesis AM plus CN right parenthesis.$

Solution

Given: BD is one of the diagonals of a quadrilateral ABCD. AM and CN are perpendiculars from A and C respectively on BD.

To Prove: ar(quad. ABCD)
$equals 1 half BD. left parenthesis AM plus CN right parenthesis$
Proof: ar(quad ABCD)

$negative ar left parenthesis increment ABD right parenthesis plus ar left parenthesis increment BCD right parenthesis equals space fraction numerator BD. AM over denominator 2 end fraction plus fraction numerator BD. CN over denominator 2 end fraction$
| $because$ Area of a triangle = $1 half$ x Base X Corresponding altitude
$equals 1 half BD left parenthesis AM plus CN right parenthesis$

Some More Questions From Constructions Chapter

In the figure, diagonals AC and BD of a trapezium ABCD with AB || CD intersect each other at O. Show that ar(Δ AOD) = ar(Δ BOC).

Parallelograms on the same base and between same parallels are equal in area. Prove this.

ABCD is a quadrilateral and BD is one of its diagonals as shown in figure. Show that ABCD is a parallelogram and find its area.

In the given figure, AB | | DC. Show that ar(BDE) = ar(ACED).

In the given figure, ABED is a parallelogram in which DE = EC. Show that area (ABF) = area (BEC)

In the following figure, ABCD is a parallelogram and EFCD is a rectangle. Also, AL ⊥ DC. Prove that
(i) ar(ABCD) = ar(EFCD)
(ii) ar(ABCD) = DC x AL.

If a triangle and a parallelogram are on the same base and between the same parallels, then prove that the area of the triangle is equal to half the area of the parallelogram.

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Constructions

ΔABC and ΔABD are two triangles on the same base AB. If line segment CD is bisected by AB at O, show that ar (ΔABC) = ar (ΔABD).

$ar left parenthesis quad. space ABCD right parenthesis equals 1 half BD. left parenthesis AM plus CN right parenthesis.$

Some More Questions From Constructions Chapter

In the figure, diagonals AC and BD of a trapezium ABCD with AB || CD intersect each other at O. Show that ar(Δ AOD) = ar(Δ BOC).

Parallelograms on the same base and between same parallels are equal in area. Prove this.

ABCD is a quadrilateral and BD is one of its diagonals as shown in figure. Show that ABCD is a parallelogram and find its area.

In the given figure, AB | | DC. Show that ar(BDE) = ar(ACED).

In the given figure, ABED is a parallelogram in which DE = EC. Show that area (ABF) = area (BEC)

In the following figure, ABCD is a parallelogram and EFCD is a rectangle. Also, AL ⊥ DC. Prove that
(i) ar(ABCD) = ar(EFCD)
(ii) ar(ABCD) = DC x AL.

If a triangle and a parallelogram are on the same base and between the same parallels, then prove that the area of the triangle is equal to half the area of the parallelogram.

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Constructions

ΔABC and ΔABD are two triangles on the same base AB. If line segment CD is bisected by AB at O, show that ar (ΔABC) = ar (ΔABD).

Some More Questions From Constructions Chapter

In the figure, diagonals AC and BD of a trapezium ABCD with AB || CD intersect each other at O. Show that ar(Δ AOD) = ar(Δ BOC).

Parallelograms on the same base and between same parallels are equal in area. Prove this.

ABCD is a quadrilateral and BD is one of its diagonals as shown in figure. Show that ABCD is a parallelogram and find its area.

In the given figure, AB | | DC. Show that ar(BDE) = ar(ACED).

In the given figure, ABED is a parallelogram in which DE = EC. Show that area (ABF) = area (BEC)

In the following figure, ABCD is a parallelogram and EFCD is a rectangle. Also, AL ⊥ DC. Prove that(i) ar(ABCD) = ar(EFCD)(ii) ar(ABCD) = DC x AL.

If a triangle and a parallelogram are on the same base and between the same parallels, then prove that the area of the triangle is equal to half the area of the parallelogram.

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

ΔABC and ΔABD are two triangles on the same base AB. If line segment CD is bisected by AB at O, show that ar (ΔABC) = ar (ΔABD).

$ar left parenthesis quad. space ABCD right parenthesis equals 1 half BD. left parenthesis AM plus CN right parenthesis.$

In the following figure, ABCD is a parallelogram and EFCD is a rectangle. Also, AL ⊥ DC. Prove that
(i) ar(ABCD) = ar(EFCD)
(ii) ar(ABCD) = DC x AL.