Constructions Scales Scales are important in everyday life. We use scales to draw maps, to construct building plans, in housing, street construction... It is impossible to draw building plans with the doors and window to size so we use scale drawings. If you are making orange squash and you mix one-part orange to four parts water, then the ratio of orange to water will be 1:4 (1 to 4). Learning Intention: By If you use 1 litre of orange, you will use 4 litres of water (1:4). the end of the lesson you will be able to If you use 2 litres of orange, you will use 8 litres of water (2:8). Understand and use a scale to interpret a If you use 10 litres of orange, you will use 40 litres of water (10:40). plan These ratios are all equivalent 1:4 = 2:8 = 10:40 Both sides of the ratio can be multiplied or divided by the same number to give an equivalent ratio. On a scale drawing, all dimensions have been reduced by the same proportion. Example A model boat is made to a scale of 1:20 (1 to 20). This scale can be applied to any units, so 1mm measured on the model is 20mm on the actual boat, 1cm measured on the model is 20cm on the actual boat, and so on. a) If the 1:20 model boat is 15cm wide, how wide is the actual boat? b) If the boat has a mast of height 4m, how high is the mast on the model?
Luigi is inspecting a house plan where 1 cm:1 m, below there is a scale floor plan of his new house. Use a ruler to measure each length of each room and using the scale ratio, give the actual lengths of the rooms in this house.
Writing Map Ratios when writing may ratios it is important that you first turn the numbers into the same unit. Then you can write the two numbers as a ratio or a fraction and use it to figure out the length, width and height of the maps. Example 3cm on a map gives a room s height of 4m. Map: Room 3: 400 3/400 Write down the map ratio and fraction of the following situations: 1. A map scale 2cm to 300m. 2. A distance of 3mm on the map represents 500m on the ground. 3. A distance of 8km is represented by 4cm on the map. 4. The scale of a map is 1 : 400, 000. If the distance between the two towns on the map is 3.8cm, find the actual distance between them. Give your answer in km.
Constructing a triangle with three given sides. Draw a triangle ABC of sides 8cm, 6cm and find the length of the third side. First draw a rough sketch. Learning Intention: By the end of the lesson you will be able to Make accurate drawings of triangles What is the length of the third side?
Draw a triangle of sides 7cm, 6cm and 5cm in the space below:
Constructing a triangle given 1 side and 2 angles. (ASA) Construct triangle ABC, where AB = 11 cm, angle A = 35, angle B = 42.
Constructing a triangle given 2 sides and 1 angle. (SAS) Construct triangle PQR where PQ = 12 cm, PR = 7 cm and angle P = 35
On the given line below, construct triangle ABC such that AB = 8 cm, BC = 8 cm and angle B = 100 0. Label your diagram. Drawing a simple scale drawing The ratio of this triangle is 1:3cm. Draw a drawing to scale in the following space: 1 2 Learning Intention: By the end of the lesson you will be able to Draw simple scale drawings. 3
The ratio of this triangle is 1:2. Draw the following triangle and find the third side of the triangle. 2 3
Learning Intention: By the end of the lesson you will be able to solve problems involving angles of elevation and depression. Angle of Elevation and Depression When you see an object above you, there's an angle of elevation between the horizontal and your line of sight to the object. when you see an object below you, there's an angle of depression between the horizontal and your line of sight to the object. The image below is a model of Aya, point A, looking up looking up to Super Girl, point S, in the sky. What is the angle of elevation from Aya to Super Girl? What is the angle of depression from Super girl to Aya?
Write down whether the angle marked with a letter in each picture is an angle of elevation or an angle of depression. iii)
The figure below shows a tree and its shadow on the ground. Let us draw a scale diagram using the given information and hence find the height of the tree. Let 1 cm in the scale diagram represent an actual distance of 1 m.
From a sixty meter tall lighthouse a boat A is observed at sea with an angle of depression of 31 o and another boat B with an angle of depression of 45 o (see the figure). The two boats and the lighthouse are in the same vertical plane. Draw a scale diagram with a scale of 1cm : 10m depicting the above information and find the distance between the boats A and B.
In a horizontal playground, Dilini is standing at the location A, 5 m away from a netball goal post. She can see the top of the goal post T, with an angle of elevation of 18 o from her eye level E. She can see the base of the goal post F, from the same position with an angle of depression of 15 o. Draw a scale diagram and find Dilini s height and the height of the goal post. When a diagram is not given, it is best to draw a sketch diagram prior to drawing the scale diagram. Sketch diagram:
Further examples 1) A person observes a rocket from a point 400 m horizontally away from the launching pad when the rocket has travelled 700 m vertically up from the launching pad. Using a scale diagram, find the angle of elevation of the rocket. 2) A ladder leaning against a wall is shown in the figure. Draw a scale diagram using the given information and find (i) the length of the ladder and (ii) the distance from the foot of the ladder to the wall.
3) A ramp for the use of wheelchairs to access a building is shown in the figure. Draw a suitable diagram using the given information and find the length of the ramp. 4) The top, P, of a clock tower situated on a horizontal ground has an angle of elevation of 60 o from a point A at the brink of a building. The angle of elevation of P from a point B in the building which is 5m directly above the point A, is 45 o (see figure on the right). Using a suitable scale diagram, find the height of the clock tower and the distance from A to the foot Q of the clock tower.
Constructing a perpendicular at a point on the line Learning Intention: By the end of the lesson you will be able to construct different diagrams. 1. Begin with line k, containing point P. 2. Place the compass on point P. Using an arbitrary radius, draw arcs intersecting line k at two points. Label the intersection points X and Y. 3. Place the compass at point X. Adjust the compass radius so that it is more than (1/2)XY. Draw an arc as shown here. 4. Without changing the compass radius, place the compass on point Y. Draw an arc intersecting the previously drawn arc. Label the intersection point A. 5. Use the straightedge to draw line AP. Line AP is perpendicular to line k.
Constructing a perpendicular from a point to a line 1. Begin with point line k and point R, not on the line. 2. Place the compass on point R. Using an arbitrary radius, draw arcs intersecting line k at two points. Label the intersection points X and Y. 3. Place the compass at point X. Adjust the compass radius so that it is more than (1/2)XY. Draw an arc as shown here. 4. Without changing the compass radius, place the compass on point Y. Draw an arc intersecting the previously drawn arc. Label the intersection point B. 5. Use the ruler to draw line RB. Line RB is perpendicular to line k.
Construct the perpendicular bisector of a line segment, or construct the midpoint of a line segment. 1. Begin with line segment XY. 2. Place the compass at point X. Adjust the compass radius so that it is more than (1/2)XY. Draw two arcs as shown here. 3. Without changing the compass radius, place the compass on point Y. Draw two arcs intersecting the previously drawn arcs. Label the intersection points A and B. 4. Using the ruler, draw line AB. Label the intersection point M. Point M is the midpoint of line segment XY, and line AB is perpendicular to line segment XY.
Construct the bisector of an angle. 1. Let point P be the vertex of the angle. Place the compass on point P and draw an arc across both sides of the angle. Label the intersection points Q and R. 2. Place the compass on point Q and draw an arc across the interior of the angle. 3. Without changing the radius of the compass, place it on point R and draw an arc intersecting the one drawn in the previous step. Label the intersection point W. 4. Using the ruler, draw line PW. This is the bisector of QPR.
Constructing a right angle (90 o ) 1). Use ruler and draw a Line segment OB of any convenient length. (as shown below) 2). Now use compass and open it to any convenient radius. And with O as center, draw an arc which cuts line segment OB at X. 3). Again use compass and opened to the same radius (as of step 2). And with X as center, draw an arc which cuts first arc at D. (as shown below) 4). Again use compass and opened to the same radius (as of step 2). And with D as center, draw another arc which cuts first arc at C. (as shown below) 5) Again use compass and opened to the same radius (as of step 2). And With C & D as center, draw two arc which cuts each other at E. 6) Join OE and extent it to A.
Construct an angle of 90o
Constructing a square or a rectangle We start with a given line segment AB> This will become one side of the square. Note. Steps 1 through 5 construct a perpendicular to line AB at the point B. This is the same construction as Constructing the perpendicular at a point on a line 1. Extend the line AB to the right. 2. Set the compasses on B and any convenient width. Scribe an arc on each side of B, creating the two points F and G. 3. With the compasses on G and any convenient width, draw an arc above the point B. 4. Without changing the compasses' width, place the compasses on F and draw an arc above B, crossing the previous arc, and creating point H
5. Draw a line from B through H. This line is perpendicular to AB, so the angle ABH is a right angle (90 ); This will become the second side of the square We now create four sides of the square the same length as AB 6. Set the compasses on A and set its width to AB. This width will be held unchanged as we create the square's other three sides. 7. Draw an arc above point A.
8. Without changing the width, move the compasses to point B. Draw an arc across BH creating point C - a vertex of the square. 9. Without changing the width, move the compasses to C. Draw an arc to the left of C across the exiting arc, creating point D - a vertex of the square. 10. Draw the lines CD and AD
Construct a rectangle ABCD in which AB = 5 cm and BC = 4 cm.
Drawing Diagonals To draw a diagonal of a shape, you must bisect the angle as shown before. Example: Draw the following rectangle in the scale of 1: 2 and draw the diagonals. How long are the diagonals? 1.5 3
Further examples 1) Construct the following triangles 2) Construct the triangle KLM that is right-angled at M, with KM= 6cm and KL= 10.5cm. Measure and write down the length of side LM and the two acute angles K and L. 3)
4) 5) Construct rectangle ABCD with AB = 8.4cm and diagonal AC= 8.9cm. Hence measure the height BC and calculate the area of ABCD. 8.9cm 8.4cm