In the diagram of the two triangles shown on the right, calculate the value of x.
8. Amy cycles 12 km north, then 16 km west. What is the shortest distance back to her starting point? Hint: draw a sketch.
9. (i) Construct an isosceles triangle with base 6 cm and other two sides 5 cm, as shown.
(ii) Draw a perpendicular line from A to the base [BC] and mark the point of intersectionD. Verify by measuring that |AD| = 4 cm.
(iii) Using Pythagoras’s theorem, calculate |BD| and |CD|. What do you notice?
(iv) Show that ∆ABD and ∆ACD are congruent.
(v) Use your answer to (iv) to describe what drawing a line from the vertex of an isosceles triangle perpendicular to its base does to the base.
B 6 cm C 5 cm 5 cm B x cm 13 cm D 35 cm C A 5 cm
34·2 Trigonometric ratios
Trigonometric ratios relate the lengths of the sides of a right-angled triangle to its interior angles.
The trigonometric ratios are only used for right-angled triangles. They are found by taking one side of a right-angled triangle and dividing it by another side. Each ratio involves only two sides of the triangle.
In order to find these ratios we must be able label the sides of the triangle properly.
The names of the sides are: ● ●
●
the hypotenuse: longest side of the triangle the opposite: side opposite the angle of interest the adjacent: side beside the angle of interest
Naming the sides of a right-angled triangle Identify the right angle, go across from it and label the side as the hypotenuse. Identify the angle of interest, go across from it and label the side as the opposite.
1
2 3
Label the last side as the adjacent. ‘Adjacent’ means ‘beside’, so it is the side beside the angle of interest.
By the end of this section you should: ● understand what trigonometric ratios are
● be able to find the sine, cosine and tangent of an angle as a ratio
● be able to find all ratios if given the value of one of the ratios
Hypotenuse Opposite A A Adjacent A Hypotenuse B Adjacent B Opposite B