### Introduction

This STEP support module includes some proofs involving triangles and sketching cubics. Previous assignments can be found here, but you can do this one without having done the others first.

STEP questions are difficult, they are supposed to be and you should expect to get stuck. However, as you tackle more and more STEP questions you will develop a range of problem solving skills (and spend less time "being stuck").

### About this assignment

The assignment is published as a pdf file below. Each STEP Support assignment module starts with a warm-up exercise, followed by preparatory work leading to a STEP question. Finally, there is a warm-down exercise.

The **warm up** for this assignment shows you how to prove some results about triangles.

The **main STEP question (1993 STEP 1 Question 7)** is about the graphs of cubics and how many roots the cubics have.

The **final question** is Euclid's proof that the base angles in an isosceles triangle are the same, known as "the pons asinorum" or "bridge of donkeys". If differs to the proof in question ** 1(i)** in that the condition $SSS$ for congruent triangles cannot be used.

More can be read about Euclid here and here . A discussion on the origins of proof including Euclid's postulates can be found on this page. When Euclid's 5th postulate doesn't hold we have an area of maths called Non-Euclidean Geometry .

Some more facts about triangles are discussed in this Maths in a minute Plus article .

### Hints, support and self evaluation

The **“Hints and partial solutions for Assignment 9”** file gives suggestions on how you can tackle the questions, and some common pitfalls to avoid, as well as some partial solutions and answers.

Here is a Worked Video Solution to the STEP question from this assignment.