Can you find the surprising 7 having a hypotenuse of ? There are 13 with a hypotenuse of and 22 for !
To show this, first in this diagram; Label the sides of one triangle a, b and h ; Keeping the sides in the same proportions, and using the angles identified as being equal, label the other sides; If you have any divisions, multiply the whole diagram by the divisors to make every side a product of a, b and h. Can you make a right-angled triangle jigsaw with more than one shape of Pythagorean triangle?
What about a jigsaw for a non-right-angled triangle? What is the largest number of right-angled triangle pieces you can fit into a triangular jigsaw? By looking at the earlier method we used to split one right-angled triangle into two, and comparing it with the diagram with three here, we can split a right-angled triangle into 4 right-angled pieces, all similar as shown here.
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Can you extend it again to five triangles? Make you own patterns using one Pythagorean triangle in a range of sizes to make a nice tiling pattern as in the previous investigation question. Send some to me at the email address at the foot of this page and I will include them here. Can you find a rectangle that dissects into different non-similar Pythagorean triangles? The Calculator earlier on this page opens in a new window is useful for the following. Find the only two Pythagorean triangles with an area equal to their perimeter. Which are the only 3 numbers that cannot be the shortest side of any Pythagorean triangle?
Can you find four consecutive numbers which are hypotenuses? What about five? Of the primitive ones in your list, what is special about their m and n values?
The Pythagorean Theorem
List the values H that are squared to make these hypotenuses: where have you seen this series before on this page? It seems there are two triples for each and every hypotenuse that is a square number H 2. One is easily explained as it has a simple relationship with the triple with hypotenuse H : what is that relationship?
For the second, look at its generators to find a proof that it always exists. Can you guess how many triples there will be with a hypotenuse that is a fourth power: H 4? Following on from the previous Puzzle, what can you find out about triples with a hypotenuse of the form H 3? What about Pythagorean triples having a smallest side which is a cube? What is special about their m and n values? How many Pythagorean triangles have a side of length 48? Find a number that can be the side of even more Pythagorean triangles. Hint: there are 5 answers less than What is the highest number of triples you can find with the same side in each?
Which number less than occurs in 32 triples? What is the smallest number that is the hypotenuse of more than one triple? What is the greatest number of triples you can find with the same hypotenuse?
Is there one that is not a multiple of 5? Find some numbers which are the odd sides of more than one primitive Pythagorean triangle. The first two are which is a side in both 8 15 17 and 15 and which is a side of both 20 21 29 and 21 Can you find a property to describe the factorizations into primes of each number in this series?
We can find a whole series of Pythagorean triples where all the numbers are palindromes : 3,4,5 33,44,55 ,, Also we have the triple ,, What is the series of factors that has been used to generate these from 3 4 5?
Another is 66,88, if we include an initial 0 in front of the hypotenuse: 66,88, What about the Pythagorean triples ,, and ,,? Can you find any more infinite series of palindromic Pythagorean triples?
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There was another one that year '05 - when was it? Assuming that the years are in this century and are just two digits long, when is the next Pythagorean Triple Date? How many other such days are there in this century? If the date is any set of 3 numbers that are a Pythagorean triple that is, the numbers need not be in order , how many dates are there in one century? How many are there in a whole day if we use a hour clock with hours from 0 to 23?
The Pythagorean Theorem
Here is another way to do this. Think of a series of numbers that are like those in the lists above, e. Plug these numbers into the Triples generator and see if any patterns emerge. Sequences need a length which is the number of non-zero numbers. Some special sums of squares What is special about the numbers 14, 29, 50, 77, if we look at them as a sum of squares? Find a formula for these numbers. It applies to the sum of K consecutive squares for all K.
To prove this, find the formula for the sum of the K squares starting at n, let's call this S n.
Area of Incircle of a Right Angled Triangle
We have not used the value of K anywhere in the proof so the recurrence relation applies no matter value K has. Look at the remainders when the odd numbers are divided by 8. There is a related pattern the evens in the list. The numbers have a remainder of 7 when divided by 8 or are 4 times a number in the list. The evens are all powers of 4 times another factor. What are those factors? The only odd numbers are 1, 3, 5, 7, 11, 15 and The evens are 2 or 6 or 14 or a power of four times one of these.
See A for the complete list of the 31 numbers and a reference to a proof. In Sprague published a proof that there are only 31 such numbers: 2, 3, 6, 7, 8, 11, 12, 15, 18, 19, 22, 23, 24, 27, 28, 31, 32, 33, 43, 44, 47, 48, 60, 67, 72, 76, 92, 96, , , See A These nine squares can be made into a rectangular jigsaw. The area of the rectangle is Cut out the nine squares and and solve the jigsaw puzzle.
Hypotenuse, opposite, and adjacent (article) | Khan Academy
What are the dimensions of the rectangle? A rectangular jigsaw puzzle has exactly nine square pieces, each of a different size, and an area of If the square pieces have sides 1, 4, 7, 8, 9, 10, 14, 15 and 18, what is the width and height of the rectangle and how do the nine pieces fit into it? According to Beiler's Recreations in the Theory of Numbers see Link and References below this is the smallest rectangular jigsaw where all the pieces are square and of different sizes. Take any two fractions or whole numbers whose product is 2 Notice that the fractions do not have to be in their lowest form :.
Give two fractions whose product is 2 : they do not have to be in lowest form. Find m,n for a: b: h:. A 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 2, 1,.. A 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, A 0, 0, 1, 1, 2, 0, 1, 1, 1, 0, 1, 2, 2, 0, 2, 1, Triad Primitive?
The sides that form the arms of the angle A are said to be adjacent to A. The side on which the triangle stands is called the base of the triangle. This can easily be seen by drawing a straight line through the angle B and parallel to the side b. The angles formed with this line are equal to A, B and C by the rule that alternate angles between parallel lines are equal.
The straight line from angle B perpendicular to the base line b is called the height of the triangle. The height is labelled h in the diagram below. You have previously learned that the area of a triangle is given by the formula. The letter G is used here to label the point where the height and the base intersect. This point is sometimes called the perpendicular projection of the point B onto the line b. Two triangles are said to be similar if all the angles of one triangle are equal to all the angles of the other. If we want to show that two triangles are similar it is sufficient to show that two angles are equal.
If two angles are equal it is obvious that the third angle in each must be equal. The triangles in the above diagram are similar. It follows that the ratios between corresponding sides are the same. In other words :. The triangles in the diagram are similar with the equal angles marked in the same way. We want to calculate the length of the sides labelled x and y. We begin by labelling the triangles so we can see more easily which sides correspond to each other.