CLASS 6 · GANITA PRAKASH · CHAPTER 8
Use a ruler and compass to explore circles, squares, rectangles, diagonals and equidistant points.
38 worked answers · short steps first, explanations when you need them
Textbook pages 188–191
Question 1
Use P as the fixed centre and set the compass opening to 4 cm.
Keeping the compass point at P, move the pencil all the way around.
A circle with centre P and radius 4 cm.
Question 2
The compass opening was set to 4 cm and stayed fixed.
Exactly 4 cm.
Question 3(a)
The diameter is AB=8 cm.
The radius is half the diameter: 8÷2.
4 cm.
Question 3(b)
X is the midpoint of AB.
AX=AB÷2=8÷2.
AX=4 cm.
Question 3(c)
The radius is half the diameter: 10÷2.
5 cm.
Question 4
Set one compass radius and keep it fixed for each matching arc.
Place the compass point at each marked centre and draw the arcs in the same direction.
Use the same compass opening and place each arc centre by a consistent spacing rule; check that paired arcs match.
Question 5
Mark two compass centres on a horizontal guide, equally spaced from the eye’s centre.
Use matching radii for the upper and lower arcs.
Place the two arc centres symmetrically about the eye’s midpoint and keep the radii equal; adjust the opening until the arcs meet as shown.
Textbook pages 192–194
Question 1
A valid name follows the corners in order around the boundary.
PQSR jumps from Q to S rather than following adjacent corners.
PQSR is not a valid name.
Question 2
A rotation preserves every side length and every angle.
So all four sides remain equal and every corner remains 90°.
No. A rotated square is still a square.
Question 3(a)
Check that its four sides are equal and its corners are right angles; the grid gives equal horizontal and vertical steps for figure A.
Figure A is a square.
Question 3(b)
Compare the grid-step lengths of the sides.
Use the grid directions to check perpendicular sides.
Yes, when the grid shows equal side vectors meeting at right angles, the equal-side and right-angle properties follow from the grid.
Question 4
For each quadrilateral, compare opposite and adjacent side lengths using grid steps.
Check that each corner is a right angle.
A square has four equal sides and four right angles; a rectangle has equal opposite sides and four right angles. Verify those properties for each drawing.
Textbook pages 195–197
Question 1
Draw PQ=6 cm.
Construct a perpendicular at P and mark S so PS=6 cm.
Construct a perpendicular at Q on the same side of PQ and mark R so QR=6 cm.
Join S to R.
PQRS is a 6 cm square: all sides are 6 cm and all four angles are 90°.
Question 2
Opposite sides of the square match PQ.
Every angle in a square is right.
RS=6 cm; ∠R=90° and ∠S=90°.
Question 3
Draw a 6 cm base.
Construct perpendicular 4 cm sides at both endpoints on the same side.
Join their endpoints.
The opposite sides are 6 cm and 4 cm, and all four angles are 90°.
Question 4
Draw a 10 cm base.
Construct perpendicular sides of 2 cm at both endpoints.
Join the two new endpoints.
The opposite sides are 10 cm and 2 cm, and each angle is 90°.
Question 5
A quadrilateral with four right angles is a rectangle.
In every rectangle, opposite sides have equal lengths.
No. Four right angles force the opposite sides to be equal.
Textbook pages 197–202
Question 1(a)
Convert 5 mm to 0.5 cm.
The horizontal separation is AB=7 cm.
The vertical difference is 3−0.5=2.5 cm.
By the right-triangle relationship, XY=√(7²+2.5²)=√55.25≈7.43 cm.
XY is about 7.4 cm.
Question 1(b)
They are at the same height, so the connecting segment is parallel to AB.
Its length equals AB.
7 cm.
Question 1(c)
The vertical difference is 4−2=2 cm.
The horizontal distance is 7 cm.
XY=√(7²+2²)=√53≈7.28 cm.
About 7.3 cm.
Question 2
The two points have the same vertical position.
Their segment is a horizontal side-to-side segment.
XY=AB, and quadrilateral ABYX is a rectangle.
Question 3
The maximum occurs when X and Y are opposite endpoints of their sides.
Those endpoint-to-endpoint distances are the rectangle’s diagonals.
The farthest XY equals a diagonal, AC or BD.
Question 4
Let each square side be s.
The rectangle is three squares placed side by side, so its dimensions are 3s by s.
Its length must be three times its breadth; for example, 12 cm by 4 cm.
Question 5
Two squares require length:breadth ratio 2:1; three require 3:1.
4:2.5 is not 2:1; 7:2 is not 3:1.
Examples: 4 cm by 2.5 cm cannot be split into two identical squares; 7 cm by 2 cm cannot be split into three identical squares.
Question 6
The square side can be 4 cm, matching the rectangle’s shorter side.
The leftover length is 8−4=4 cm, split equally as 2 cm on each end.
Use a 4 cm square with 2 cm margins at each end of the 8 cm side.
Question 7
Choose a side length for the outer square.
Mark equal distances along its sides and join corresponding marks to form smaller squares.
Repeat with smaller equal offsets.
Many constructions are possible; keep corresponding offsets equal so each inner boundary is a square.
Question 8
Draw the two diagonals of the square.
Their intersection is the square’s centre.
Place the circle’s centre at the intersection of the square’s diagonals.
Question 9
Mark the midpoint of each side.
Use the opposite corner or an appropriate corner point as each arc centre; set the compass opening to the corner-to-side midpoint distance.
Repeat with the same radius for each side.
Use matching compass radii and symmetrically placed centres so the four arcs bulge equally; verify on a rough construction first.
Textbook pages 203–211
Question 1
Construct both diagonals and measure them.
A rectangle’s diagonals have the same length.
The two diagonals are equal in length.
Question 2
If the diagonal splits each 90° corner into 45° and 45°, the adjacent sides form equal legs of a right triangle.
So the rectangle’s adjacent sides are equal.
When the rectangle is a square.
Question 3
Draw a base AB.
At A construct a 90° side and a diagonal ray making 50° with the base (the remaining angle is 40°).
Choose a point C on the ray; construct perpendiculars at B and C to meet at D.
Complete the rectangle using perpendicular sides; the diagonal splits the opposite corners into 50° and 40°.
Question 4
Equal halves of a 90° angle mean the diagonal makes an isosceles right triangle.
The adjacent sides are equal.
The rectangle is a square.
Question 5
The side and unknown side are perpendicular legs; the diagonal is the hypotenuse.
Other side=√(8²−4²)=√48=4√3≈6.93 cm.
The other side is 4√3 cm, about 6.9 cm.
Question 6
Other side=√(7²−3²)=√40=2√10≈6.32 cm.
The other side is 2√10 cm, about 6.3 cm.
Textbook pages 211–216
Question 1
Draw an arc of radius 5 cm centred at B.
Draw another arc of radius 5 cm centred at C.
Choose an intersection of the arcs.
At either intersection, AB=AC=5 cm.
Question 2
With compass radius 5 cm, draw arcs centred at B and C.
Their upper intersection is A.
Join A to B and A to C; draw the remaining sides and the matching arc as shown.
Use the upper intersection of the two 5 cm arcs for A; then complete the walls and roof arcs with the given measurements.
Question 3
Use radius 7 cm for the equal-distance arcs.
Set each straight boundary segment to 7 cm.
Repeat the construction with 7 cm lengths and a 7 cm compass radius.
Question 4
Draw a 5 cm segment BC.
Draw circles of radius 5 cm centred at B and C.
Use their two intersection points as A and D, then join A-B-D-C in order.
Yes. This constructs a rhombus; choose the intersections that make 60° and 120° angles, so it is not a square.