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Pythagorean Triples

How many right-angled triangles are there with sides that are all integers less than 100 units?

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Square Pegs

Which is a better fit, a square peg in a round hole or a round peg in a square hole?

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The Old Goats

A rectangular field has two posts with a ring on top of each post. There are two quarrelsome goats and plenty of ropes which you can tie to their collars. How can you secure them so they can't fight each other but can reach every corner of the field?


Stage: 3 Challenge Level: Challenge Level:1

A simplified diagram of the situation reveals that the problem is about similar triangles.

(i) Since the two triangles are similar, the fraction $0.9/y = 5/15$

Rearranging the equality gives $y$ = $2.7$ metres.

So the minimum height from which the ball can be hit from is $2.7$ metres.

(ii) Using Pythagoras's Theorem $(a^2 + b^2 = c^2)$ we can work out how far the ball travels before it hits the ground.

$2.7^2 + 15^2 = x^2$

$x = \sqrt{232.29}$

$x = 15.24$ metres

So the distance travelled by the ball before it hits the ground is $15.24$ metres.

(iii) For the ball to travel the shortest distance (and hence the shortest time) it should be hit so that it bounces on the base line, however this serve would be a fault. To give the receiver the least time to return the serve, the ball would have to bounce on the service line. In this situation the height to which the ball rises after it has bounced will be a minimum.

How far would the ball travel before it was hit back?