Constructing triangles

I've always wondered how far it is to that lighthouse on the island. We can find out! What, will you swim out with a tape measure, or what...? No. We can calculate it by drawing a triangle.

Draw a triangle? How? Lina will draw a triangle to find out the distance to the lighthouse. But what sort of triangle? Let’s find out.

The first side of her triangle is the distance between Maria's house and the gazebo. Lina measures the distance. She gets 180 metres. On a piece of paper, she draws the first side of the triangle. The straight line she draws is 18 centimetres long.

One centimetre on the paper for 10 metres in reality. Maria's house is at one end of the line, and the gazebo is at the other. Now Lina takes out a protractor. She places it so that zero degrees on the protractor points along the line between the house and the gazebo. Then she tries, as closely as she can, to see what angle it is to the lighthouse.

She gets... 85 degrees. That is the first angle of the triangle. She draws a new line on the paper that forms the angle 85 degrees with the first line she drew. The new line is part of the second side of the triangle.

Then Lina does the same at Maria's house. She places the protractor so that zero degrees on the protractor points along the line between the house and the gazebo. Then she tries again to see what angle it is to the lighthouse. Now she gets... 75 degrees.

This is the second angle of the triangle. Lina draws a third line on the paper at an angle of 75 degrees to the first line. The third line is the last side of the triangle. She draws the second and third lines until they meet at a point. At that point lies the lighthouse!

Now Lina has constructed a triangle by measuring one side and two angles in the triangle. Do you remember what she plans to do? She plans to calculate the distance between Maria's house and the lighthouse. Therefore, she measures the length of the distance on the paper. The distance between the lighthouse and Maria's house is about 52 centimetres.

Every centimetre of the paper corresponds to 10 metres, so that means it is about 520 metres to the lighthouse from Maria's house. From the gazebo to the lighthouse is 51 centimetres on the paper, ie about 510 metres in reality. Lina has used a triangle where she knows the length of one side and two angles to calculate the other two sides of the triangle. Knowing three things about the triangle was enough for Lina to construct it. But it doesn’t have to be two angles and one side.

A different triangle has sides that are 7 cm, 5 cm and 6 cm long Now let's draw it. We start by drawing the line that is 6 cm. We call one end point A, and the other one B. Now: Put the tip of a compass on point A. We want to find a side that is 5 cm.

Set the compass on the radius 5 centimetres. Draw an arc. Then, put the tip of the compass on point B. Set the radius to 7 centimetres. Draw an arc.

The arcs cross each other here, and that is the point of the last corner, C, in the triangle. Now you have three points - A, B and C - that are corners of the triangle. Draw the sides. The triangle is complete. It’s cool that you can calculate distance with the help of a triangle.

Yes, and that’s how maps were drawn in the past, before there were aircraft and satellites. Let’s dive in and check!