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### Number and algebra

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# Tied Up

A field is in the shape of a right angled triangle.

At the midpoint of each side is a post. Horace the goat is tied to the midpoint of the hypotenuse.

The two sheep, Sid and Sadie, are tied to the midpoints of the other two sides.

Each rope is just exactly long enough to allow the animal to reach the two vertices on the side to which it is tied.

Which animal can graze the largest area of grass?

What proportion of the field is it able to graze?

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ShortChallenge Level

- Problem
- Solutions

A field is in the shape of a right angled triangle.

At the midpoint of each side is a post. Horace the goat is tied to the midpoint of the hypotenuse.

The two sheep, Sid and Sadie, are tied to the midpoints of the other two sides.

Each rope is just exactly long enough to allow the animal to reach the two vertices on the side to which it is tied.

Which animal can graze the largest area of grass?

What proportion of the field is it able to graze?

You can find more short problems, arranged by curriculum topic, in our short problems collection.

A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

M is any point on the line AB. Squares of side length AM and MB are constructed and their circumcircles intersect at P (and M). Prove that the lines AD and BE produced pass through P.