6.5 Angle Bisection for a Design

6.5 Angle Bisection for a Design

Look at the 8-petal flower design in Fig. 6.5. The supporting lines of this figure are 8 lines through a common centre — the full 360° around the centre is divided into 8 equal parts, so every pair of adjacent lines makes an angle of 360° ÷ 8 = 45°. How do we construct such a figure?

We need the angle between every pair of adjacent lines to be equal. Since 360° is equally divisible into 8 parts, every angle is 45°. How do we construct a 45° angle? If we can divide it into two equal angles, we can use that to construct a regular hexagon.

How do we bisect an angle using ruler and compass?

Consider an angle ∠XOY. We can bisect it if we can draw two congruent triangles, △OBC and △OAC as shown in the figure. Then ∠BOC = ∠AOC.

How do we construct these congruent triangles, given the angle? If A and B are marked such that OA = OB, and if C is chosen such that BC = AC, then by the SSS congruence condition, △OBC ≅ △OAC. So we can bisect an angle as follows.

Steps for Angle Bisection

1. Mark points A and B so that OA = OB. (Draw an arc with centre O cutting both rays.)
2. Choosing any sufficiently long radius, cut arcs from A and B, keeping the radius same. Mark the point of intersection as C.
3. OC bisects ∠AOB.

Since 360° is equally divisible into 8 parts, every angle is 45°. We can construct a 45° angle by first constructing a 90° angle and then bisecting it. So, a 45° angle can be constructed by first constructing a 90° angle and then bisecting it.

6.6 Construction of a Line Parallel to a Given Line

Recall that in the previous grades, using a ruler and a set square, we constructed equal corresponding angles to get parallel lines. How do we implement this idea using a ruler and a compass?

Suppose there is a line m to which we need to draw a parallel line. We construct a line l that intersects m. Line l will serve as a transversal to line m and the line parallel to m that we are going to construct. Let us choose a point B on l through which we are going to draw the parallel line. This parallel line must make the same corresponding angle, as shown in the figure. This can be done by copying the angle between m and l.

Here is a step-by-step procedure (Figs. 6.7–6.10):

1. Draw line m. Draw transversal l meeting m at a point (A below the intersection on m, call this intersection). Mark a point B on l above A through which we want the parallel line.
2. Construct arcs of equal radius centred at A and at B, crossing l.
3. This arc cuts line l and m, giving points C and D (near A) and E (near B on l).
4. Transfer the length CD (chord of angle at A) onto the arc from B, marking point F. The line through B and F is parallel to m.

6.7 Arch Designs

Trefoil Arch

Have you seen this kind of beautiful arch? Imagine the entrance arches of the Diwan-i-Aam, Red Fort (Delhi) or Central Park, New York. The first step is to be able to draw them on a plane surface such as paper or stone. Construct this arch shape on a piece of paper. Let us think about the support lines this figure will need.

For symmetry, we should have AB = CD, and ∠BAD = ∠CDA. How would you construct these support lines? Construct equal angles at A and D. Mark B and C such that AB = CD. Then use these support lines to construct an arch. If required, adjust the radii of the arcs to make the arch look more aesthetically pleasing.

A Pointed Arch

Some arches look pointed — like those of the Diwan-i-Aam in the Red Fort, Delhi. How do we construct this shape? Use 'Wavy–Wave' style supporting lines from the Grade 6 textbook: the supporting lines are just two line segments of equal length. If their midpoints are marked, will you be able to construct a pointed arch?

6.8 Regular Hexagons & Equilateral Triangles

Recall that a regular polygon has equal sides and equal angles. A regular polygon with 5 sides is called an equilateral triangle with 3 sides, a regular pentagon with 5 sides and a regular hexagon with 6 sides. We have constructed these figures earlier.

How do we construct a regular pentagon (5-sided figure) and a regular hexagon (6-sided figure)? To begin with, try to construct a pentagon and hexagon with equal sidelengths.

To construct a regular pentagon, we first need to have a better understanding of the angles of triangles and equilateral triangles. We will discuss this in later years. Constructing a regular hexagon is within our reach! Can we break a regular hexagon into smaller pieces that can be constructed?

Regular Hexagon and Equilateral Triangles

What happens when we join the 'opposite' points of a regular hexagon? We get a regular hexagon has equal sides and equal angles. Will all the triangles in the figure be equilateral? To answer these questions, we will reverse our approach. Can 6 congruent equilateral triangles be placed together to form a regular hexagon?

If six congruent equilateral triangles can be placed as shown in Fig. 6.12, then the sides of the resulting hexagon are equal, and their angles are 60° + 60° = 120° (how?). So what we really need to construct is 6 congruent equilateral triangles if we have a rigid method to draw a 60° angle.

Construction of a 60° Angle

How do we construct a 60° angle? We get a 60° angle if we construct an equilateral triangle! We can use the following steps for this. Suppose we need a 60° angle at point A on a line segment AX.

Step 1

Construct an arc with centre A and any radius.

Step 2

With the same radius, cut another arc from B that meets the first arc. Let C be the point at which the arcs meet. We have ∠CAX = 60°.

This is because △ABC is equilateral (all sides equal to the common radius), so ∠CAB = 60°. Using these ideas we can construct a regular hexagon. We can construct a 120° angle more directly if we can construct a 120° angle — using a ruler and a compass. "If we construct this, we also get this!" — and 60° + 120° = 180°.

6.9 Repeating Units and Repeating Angles — Copying an Angle

Construct the 4-leaf design made from repeating a single curved unit in four different orientations (Fig. 6.4). To construct this figure, we need to make exact copies of this unit in two different orientations. In this figure, there is a single unit repeating itself. To construct this, we need to make exact copies of this unit in two different orientations. In order to make exact copies, all the units must have the same arm lengths and the same angle between the arms. We can ensure equal arm lengths — but how do we ensure equal angles? Let us develop a method to construct an exact copy of a given angle.

Steps of Construction to Copy an Angle (Fig. copy-angle)

1. Given ∠BAC (at A) and a ray XZ. Draw an arc of radius AB from centre A cutting both arms at points B and C. This gives the isosceles triangle △ABC with AB = AC.
2. Draw an arc of the same radius from X. Let it meet ray XZ at Z.
3. Measure BC with the compass. Transfer this length to the arc from Z to get point Y such that YZ = BC.
4. Join X to Y. By the SSS congruence condition, △ABC ≅ △XYZ. So ∠YXZ = ∠BAC — the angle is copied.