DMA - 4. Geometry

DMA

CHAPTER - 4

GEOMETRY

Learning Outcomes

Understand Basic Geometric Concepts

Identify and describe basic geometric shapes like lines, angles, triangles, circles, and polygons.
Understand properties of parallel, perpendicular, and intersecting lines in maritime contexts.

Measure and Calculate Geometric Figures

Calculate the perimeter, area, and volume of common shapes used in ship design and cargo calculations.
Use formulas to solve problems involving circles, rectangles, and triangles (e.g., calculating deck areas or cargo space).

Apply Geometry in Navigation

Use geometric principles to understand and solve problems related to angles, bearings, and courses.
Calculate distances and positions using triangulation and bearings on nautical charts.

Understand and Use Coordinates

Plot and interpret points on a coordinate system, especially when working with navigational charts.
Use coordinates to determine positions and navigate routes.

Use Geometry in Ship Stability and Design

Apply geometric understanding to calculate center of gravity and buoyancy.
Understand how the shape and design of a vessel affect its stability and balance.

Visualize and Interpret Geometric Data

Interpret diagrams and drawings related to ship construction and navigation.
Use geometric reasoning to visualize ship layouts and cargo arrangements.

Construction

4.1 To construct a line to bisect a given angle

Given ∠ A. With centre A and any radius, draw an arc cutting both arms of ∠ A at B and C. With centres B and C and the same radius in each case, draw two arcs to cut at X. Then AX bisects ∠ A.

4.2 To construct a line to bisect a given line

Given AB. With centres A and B with equal radii draw arcs cutting at X and Y .Then XY bisects AB. Furthermore , XY ⊥ AB.

4.3 To construct a perpendicular to a given line form a point not on the line.

Given AB with point P not on AB. With centre P and any radius, draw an arc cutting AB at two points X and Y. With centres X and Y and equal radii , draw two arcs at Q. Then PQ is perpendicular to AB.

4.4 To construct a perpendicular to a given line at a point on the line.

Given AB with point P on it. With centre P and any radius, draw two arcs of the same circle cutting AB on opposite sides of P at X and Y. With centres X and Y and with the same radius, draw two arcs cutting at Z. Then ZP is perpendicular to AB at P.

4.5 To construct an angle equal to a given angle

Given ∠ A and line PT. To construct ∠ RPT equal to ∠ A. With centre A and any radius r, draw an arc BC cutting both arms of ∠ A. With centre P and radius r, draw an arc cutting PT at Q. With centre Q and radius BC ,draw an arc to cut the first arc at R. Join PR. Then ∠ RPT = ∠ A.

4.6 To construct through a given point a line parallel to given line

Given AB and point P. Through P, draw PX marking ∠ AXP with AB. Then on PX, Construct ∠ XPQ equal to ∠ AXP. Then PQ is parallel to AB.

4.7 To divide a given line into a given number of equal parts

Given AB. Suppose it has to be divided into three equal parts. Draw through A any line AS. With compasses mark off AP = PQ = QR along AS. Join RB. Construct QD and PC parallel to RB. Then AB is divided into three equal parts. This construction holds for any number parts.

4.8 To construct a tangent to a circle through a given point on the circle.

Given circle with centre O and point P on it. Join OP and produce to Q. Draw the line PT perpendicular to OQ at P. Then PT is the required tangent.

4.9 To construct two tangents to a given circle from a given external point

Given circle with centre O, and an external point P. Join OP .On OP as diameter draw a circle to the cut given circle at R and S. ∠ ORP and ∠ OSP will be right angle. Then PR and PS are required tangents.

4.10 To construct the circum-circle of a given triangle

Given ΔABC. Construct perpendicular bisectors of AB and BC. Let these perpendicular bisectors meet at O. This is the circum centre of Δ ABC. Draw the circle O and radius AO.

4.11 To constructs the in-circle of a given triangle

Given ΔABC. Construct the bisectors of ∠A and ∠B. Let these angle bisectors meet at I. This is the incentre of ΔABC. Draw the circle with centre I touching AB,BC and AC.

Exercises 4.1

(1) Construct a parallelogram ABCD with AB = 6 cm, BC = 9 cm and ∠ ABC = 45˚.

(2) Construct a right-angled triangle in which the hypotenuse is 6 cm and one of the other sides is 2.5 cm

(3) Construct a triangle having sides 9 cm, 6.5 cm and 7 cm.

(4) Construct a triangle ABC where AB = 7.5 cm, BC = 9 cm. AC = 8.5 cm and by construction, find point P inside the triangle which is 5.5 cm from A and equidistant from AB and BC. Find the length of BP by measurement.

(5) Draw angle XOY = 55˚. Construct a circle radius 5 cm, to touch OX and OY.

(6) Construct a circle to touch two parallel straight lines and to touch also a transversal of the parallel lines.

(7) Draw a circle of radius 3 cm. Construct two tangents to this circle with an angle of 65˚ between them.

(8) Construct an isosceles triangle on a base of 4.5 cm, given the sum of the equal sides as 12 cm.

(9) Draw a circle of radius 6 cm touching a line. Construct another circle of radius 3 cm to touch the line and also the first circle externally.

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