Drawing a triangle

P Q is one side of triangle P Q R on the grid below. Angle P Q R is a right angle and Q R is the same length as P Q. Draw and letter triangle P Q R. Only P and Q are on the paper. R is the vertex we have to find. Start by counting how P gets to Q along the ruled lines. 3 squares right. Then 1 square up. So P to Q is 3 right and 1 up, and that pair of counts is the whole of the side P Q. The question says the corner at Q is a right angle, so Q R has to leave Q square to P Q. Take the staircase from P to Q and turn it a quarter turn about Q. It lands on the other side of Q, and the 3 and the 1 have swapped over. From Q, 1 square left. Then 3 squares up. The whole staircase turned, so the line out of Q turned with it. A quarter turn is a right angle. The right angle told us which way Q R goes. It did not tell us how far. Q R is the same length as P Q, and the turned staircase carries the same 2 counts, 1 and 3. The same 2 counts make the same length, so R is at the far end of that staircase, 1 square left of Q and 3 squares up. Join P to R, and triangle P Q R is drawn. Now put the question's own facts onto the drawing. A small square at Q says the angle there is a right angle. One tick on P Q and one tick on Q R say those 2 sides are equal. Without those 2 marks it is 3 lines on a grid, and it says nothing about what was asked for.

Your turn, and this time there are no ruled squares to count. Draw triangle A B C. A B and B C are both 6 centimetres. Angle A B C is one hundred and twenty five degrees. Try it first. Pause if you like. [Mathweave logo sounds] Step 1 is the side the question gives a length for. Draw A B 6 centimetres long. Now the direction. Draw a line out from B, opened until the angle it makes with B A is one hundred and twenty five degrees. That line says which way C lies from B. It does not say how far along C sits. Every point on this arc is 6 centimetres from B. C is where the arc crosses the line. Join A to C, and triangle A B C is drawn. A B and B C are both 6 centimetres, so one tick on each says those 2 sides are equal. Now read the drawing back against the question. 6 centimetres. And one hundred and twenty five degrees. Every fact the question gave is on the triangle. The angle at B is the direction the second side leaves in. The arc is every point 6 centimetres from B, and C is the one of them on that line. A construction is a direction and then a distance.