Q.1. P and R are point on side CD of a parallelogram ABCD . Show that (i). ar (triangle PAB) = ar(triangle RAB) (ii). aIf ar (triangle PAB) = 8 cm square, find ar (ABCD) Q.2. Prove that equal chords of two congruent circles subtend equal angles at their centers . Q.3. Construct a square whose each side measure 5 cm . Q.4. ABCD is a parallelogram in which diagonal BD bisects angle B . Show that BD bisects angle D also . Q.5. A 4 cm wooden cube is cut into 1 cm cubes . What is the percentage increase in the surface area after such cutting ?
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Q1 :
and 
have same base 
.
Since they lie between parallel sides, the height is also same.
Consequently areas will be same.
Since they lie between parallel sides, the height is also same.
Consequently areas will be same.
Answered by
3
see diagram.
Take AB as the base of the parallelogram.
Then draw perpendiculars from P and R on to AB. Since AB || CD, the two perpendiculars are parallel and equal to the altitude of the parallelogram with AB as the base.
The area of triangle PAB = area of RAB = 1/2 * AB * altitude.
Area of parallelogram ABCD = AB * altitude = twice that of the triangles.
= 16 cmsquare.
==========================
see diagram.
Take a circle of radius R. Two congruent circles mean that their radii are same. So we can work with one circle itself. So I draw two chords AB and DE of equal length L. C is the center of circle.
Look at the triangles CAB and CDE. The sides AB = DE, given. Then CA = CB = CD = CE = radius of the triangle. Hence the two triangles are congruent. Hence angles ACB and angle DCE are same. So the angles subtended are same.
=======================
3.
Draw a line horizontally in the middle of paper sheet, and measure 5 cm using a ruler. Mark the end points as A and B. Now using the perpendicular bisector method using compass or by using protractor draw a line AD at A and a line BC at B , both perpendicular to the line AB. Use the ruler or the compass measuring 5 cm, to cut off these BC and AD at 5 cm. Now join CD.
you have the square.
=============================
4 In parallelogram ABCD, the diagonal BD bisects angle B. Means that
angle ABD = angle DBC.
AD || BC. and BD is a transversal line cutting the two parallel lines.
Hence, we have angle ADB = angle DBC as they are alternate angles.
also angle ABD = angle BDC as they are alternate angles.
Since angle ABD = angle DBC, the other two angles ADB = angle BDC.
So BD bisects angle D also.
====================
5
cube edge length = 4 cm
its total surface are a = 6 * 4² = 96 cm²
volume of the cube = 4³ = 64 cm³
volume of small cubes = 1³ cm³
number of small cubes made from the big cube : 64 cm³/1 cm³ = 64
surface area of all 64 small cubes : 64 * 6 * 1² = 384 cm²
increase in surface area = 384 - 96 = 288 cm²
% increase = 288/96 * 100 = 300%
Take AB as the base of the parallelogram.
Then draw perpendiculars from P and R on to AB. Since AB || CD, the two perpendiculars are parallel and equal to the altitude of the parallelogram with AB as the base.
The area of triangle PAB = area of RAB = 1/2 * AB * altitude.
Area of parallelogram ABCD = AB * altitude = twice that of the triangles.
= 16 cmsquare.
==========================
see diagram.
Take a circle of radius R. Two congruent circles mean that their radii are same. So we can work with one circle itself. So I draw two chords AB and DE of equal length L. C is the center of circle.
Look at the triangles CAB and CDE. The sides AB = DE, given. Then CA = CB = CD = CE = radius of the triangle. Hence the two triangles are congruent. Hence angles ACB and angle DCE are same. So the angles subtended are same.
=======================
3.
Draw a line horizontally in the middle of paper sheet, and measure 5 cm using a ruler. Mark the end points as A and B. Now using the perpendicular bisector method using compass or by using protractor draw a line AD at A and a line BC at B , both perpendicular to the line AB. Use the ruler or the compass measuring 5 cm, to cut off these BC and AD at 5 cm. Now join CD.
you have the square.
=============================
4 In parallelogram ABCD, the diagonal BD bisects angle B. Means that
angle ABD = angle DBC.
AD || BC. and BD is a transversal line cutting the two parallel lines.
Hence, we have angle ADB = angle DBC as they are alternate angles.
also angle ABD = angle BDC as they are alternate angles.
Since angle ABD = angle DBC, the other two angles ADB = angle BDC.
So BD bisects angle D also.
====================
5
cube edge length = 4 cm
its total surface are a = 6 * 4² = 96 cm²
volume of the cube = 4³ = 64 cm³
volume of small cubes = 1³ cm³
number of small cubes made from the big cube : 64 cm³/1 cm³ = 64
surface area of all 64 small cubes : 64 * 6 * 1² = 384 cm²
increase in surface area = 384 - 96 = 288 cm²
% increase = 288/96 * 100 = 300%
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