Following on from the previous post, some of you were asking yesterday how we could find out what the sides needed to be to create isoscles triangles that had two angles of 75o and one 30o. To do this, we need to learn some much more challenging (but very interesting) maths.

We could draw the triangle using the perpendicular line method (or the compass and ruler) and then measure the angles we had drawn.When we did this, we measured the equal angles as 47o and the third angle as 86o.

Similarly we could draw a baseline of 7 cm (or 5cm) and then draw two lines that are at 75o and find the length of the two sides required by measuring to the point where they join the perpendicular bisector at 15o (half the required third angle).

When we did this, we discovered that the sides (x) actually needed to be 13.5 cm for the angles to be as required and the baseline to be 7cm.

The problem with drawing to find the angles is that we may sometimes be imprecise in our drawing (or measuring) and this could lead to inaccuracies: it is very difficult to be absolutely precise with a pencil, ruler and a protractor to the level we would like. Fortunately, there is another mathematical way to calculate the sides given the angles once you have a right-angled triangle (and to check our drawing). Since any triangle can be divided to create a right-angled triangle (and we have done so with the isosceles triangle to create two identical (congruent) triangles), we can use this maths to investigate further...

Here, the right-angled triangle is half of the triangle above.

The side that measures is the hypotenuse of the right-angled triangle.

The length of this side can be found using TRIGONOMETRY which is a really useful and interesting branch of maths. You need a calculator to be able to work this out.

The cosine of angle 75o = the length of the adjacent side (3.5 cm) divided by the length of the hypotenuse. So, by rearranging the equation, we can also work out

by dividing 3.5 by Cos 75o.

So = 3.5 0.2588190451 = 13.523 cm (which is pretty close to our drawn answer of 13.5 cm).

Using similar maths, we could use the length of the sides to work out the angles required to draw an isosceles triangle with sides that have one 7 cm and two of 5 cm. Here the angle at C would be the inverse cosine of 3.5 divided by 7 (which is the inverse cosine of 0.7) = 45.57o. This is close to our original drawing, but shows we were further out using the drawing of lines and measuring of angles method than we were measuring the angles and drawing the lines which is interesting!

So we definitely know now that we can't draw an isosceles triangle that has sides of 7cm and 5cm and angles of 75o and 30o accurately! Great questioning and exploring Year 6!