### Kissing Triangles

Determine the total shaded area of the 'kissing triangles'.

### Trice

ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP : PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED. What is the area of the triangle PQR?

Four rods, two of length a and two of length b, are linked to form a kite. The linkage is moveable so that the angles change. What is the maximum area of the kite?

# Constructing Triangles

##### Stage: 3 Challenge Level:

Eugena from City of London School for Girls made a good start to this problem:

Say you roll three dice and you end up with the numbers 6, 3 and 1.
You draw the 6cm line and then with a compass and ruler you measure 3cm.
After you draw the arc, you repeat the same step, the only change being that
instead of measuring 3cm you measure 1cm.
Draw the arc and you'll see that the two arcs will not intercept.
The three numbers didn't work...

Say you roll three dice and you end up with the numbers 5, 3 and 6.
You draw the 5cm line and then with a compass and ruler you measure 3cm.
After you draw the arc, you reapeat the same step, the only change being that
instead of measuring 3cm you measure 6cm.
Draw the arc and you'll see that this time the arcs intercept.
The three numbers worked!

The 5/6 Maths Extension (MEP) group from Lumen Christi School in Australia also trialled some side lengths and made the following observation:

We discovered sets of lengths for which we were able and unable to make triangles. The ones that worked were the ones where the two shorter lengths added up to more than the longest length. For the ones that didn’t work we found that the two smaller lengths didn't add up to the length of the longest side.

What we weren't sure about was what if the lengths of the two shorter sides add up to the longest length. We tested that and found out that they didn’t make a triangle. Why it is so: If the two smaller lengths of the triangle add up to the length of the longest side, the two sides intersected only when they met at the line. So it didn’t make a triangle.

Well done for spotting this special case and for considering it very carefully!

Sam, Yordan, Isla from Kings Ely, Elijah from Reading School and Amy from Melbourn Village Collage formulated a quick way to tell if it will be possible to construct a triangle from the three numbers generated. Yordan says:

Triangles can only be constructed if the sum of both smallest sides is greater than the longest side.
If $s$ and $c$ were the shorter sides, and $l$ was the longest, you could only construct a triangle if:

$$l < s + c\;.$$

Members of the Senior Maths Challenge Group at Lyneham Primary School in Australia came to the same conclusion. They added:

If you roll a '1', the others will have to be the same.

Amy also remarked:

It’s not possible to draw more than triangle from each set of measurements; even if it looks like you have you have just flipped or rotated it.

Well done to everyone!

Susan claims that she used the spreadsheet here to decide whether you are more likely to win or to lose in the one player game. Do you see how?