Ratio and Proportion Introductory Trigonometry

2 Finding the height of something very tall can be easily determined by setting up proportions of similar right triangles. For example, a building cast a 60 metre shadow and a 5 metre flagpole cast a shadow of 6 metres during the same time during the day. Let’s think of them as 2 right triangles and look at the sides as proportional: y/60 = 5/6. You can solve the problem like any other proportion. 300 = 6y 50 = y The height of the building is 50 metres and you did not even have to measure it with a tape measure. Understanding what is a proportion and what is not, helps to solve problems.

Introductory Trigonometry3 In ancient Egypt, we know, that the Pharaoh employed land surveyors, so called harpedonaptai or harpedonapts (rope stretchers or rope-knotters). After the annual flooding of the Nile river, when rich silt was deposited on the valley floor, the property boundaries of the fields were destroyed, so they had to be re-established. The harpedonaptai used measuring cords with 11 equally spaced knots (or marked with paint) dividing it into 12 parts. With these ropes they could form a right triangle with the sides of 3 : 4 : 5.

Introductory Trigonometry4 The triangle shown on the left above has its sides in the ratio 3 to 4 to 5. Any triangle with its sides in this ratio is a right triangle. An observer at the top of a 40-metre vertical tower knows that the base of the tower is 30 metres from a target on the ground. Calculate his slant range (direct line of sight) from the target. Slant Range AB (in metres) = ?

Introductory Trigonometry5 The triangle shown on the left above has its sides in the ratio 3 to 4 to 5. Any triangle with its sides in this ratio is a right triangle. An observer at the top of a 40-metre vertical tower knows that the base of the tower is 30 metres from a target on the ground. Calculate his slant range (direct line of sight) from the target. Slant Range AB (in metres) = 50

Introductory Trigonometry6 The triangle shown on the left above has its sides in the ratio 3 to 4 to 5. Any triangle with its sides in this ratio is a right triangle. A wall frame is 3.2 metres wide and 2.4 metres high. What must be the diagonal length to ensure everything is “square”? x 3200 mm 2400 mm Diagonal =?metres

Introductory Trigonometry7 The triangle shown on the left above has its sides in the ratio 3 to 4 to 5. Any triangle with its sides in this ratio is a right triangle. A wall frame is 3.2 metres wide and 2.4 metres high. What must be the diagonal length to ensure everything is “square”? x 3200 mm 2400 mm Diagonal = 4 metres (or mm) 4 x 800 mm 5 x 800 mm 3 x 800 mm

Introductory Trigonometry8 All 30 o / 60 o right-angled triangles are similar with their sides in the ratio 1 (side opposite 30 o ) : 1.73 (side adjacent to 30 o ) : 2 (hypotenuse).

Introductory Trigonometry9 A ramp angled at 30 o to the horizontal is to be made so that its height is 1.2 metres. What is the slant height of this ramp? Slant Height = ? metres

Introductory Trigonometry10 A ramp angled at 30 o to the horizontal is to be made so that its height is 1.2 metres. What is the slant height of this ramp? Slant Height = 1.2 x 2 m = 2.4 m

Introductory Trigonometry11 A ramp angled at 30 o to the horizontal is to be made so that its height is 1.2 metres. At what horizontal distance is the foot of the ramp from the base of the 1.2 metre upright? Horizontal Distance = ? mm 1.73

Introductory Trigonometry12 A ramp angled at 30 o to the horizontal is to be made so that its height is 1.2 metres. At what horizontal distance is the foot of the ramp from the base of the 1.2 metre upright? Horizontal Distance = mm x 1.73 = mm 1.73

Introductory Trigonometry metres along the tree’s shadow to a position where the end of her shadow exactly overlaps the end of the tree’s shadow. She is now 6.10 metres from the end of the shadows. How tall is the tree? ? m Juanita, who is 1.82 metres tall, wants to find the height of a tree in her backyard. From the tree’s base, she walks

Introductory Trigonometry metres along the tree’s shadow to a position where the end of her shadow exactly overlaps the end of the tree’s shadow. She is now 6.10 metres from the end of the shadows. How tall is the tree? x : = 1.82 : 6.10x = 5.46 m Juanita, who is 1.82 metres tall, wants to find the height of a tree in her backyard. From the tree’s base, she walks x

Introductory Trigonometry15 ____ m

Introductory Trigonometry16 H : 240 = 6.2 : 10 H = 0.62 x 240 H = m

Introductory Trigonometry17 Use the information given in the diagram above and your knowledge of similar right- angled triangles to find PR, the distance across the river. ?metres

Introductory Trigonometry18 Use the information given in the diagram above and your knowledge of similar right- angled triangles to find PR, the distance across the river. PR : 60 = (PR + 45) : 90 PR = 90 metres

Introductory Trigonometry19 References / Sources: d_proportion.htmhttp:// d_proportion.htm alignment.htmlhttp:// alignment.html /Lesson_11.3.pdfhttp:// /Lesson_11.3.pdf Galilei, Galileo ( ) [The universe] cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Opere Il Saggiatore p. 171.