MATH HELP! Of triangles having sides with lengths as follows, which are right triangles?

Question

a) 33, 55, 44

b) 120, 130, 50

I looked up the answers and I know both of them ARE right triangles, but I have no idea why.

Answer 1

Both are.

You can do the math of the Pythagorean Theorem on the numbers provided, but there is an easier way.

Reduce them by dividing each number by their GCF's: 11 for the first set and 10 for the second set. You get 3,5,4 and 12,13,5. In usual order, they are: 3,4,5 and 5,12,13. Now, you will likely recognize those as being Pythagorean Triples, nifty integer number sets that are right triangles. But that won't prove it to a teacher (!) so:

3^2 + 4^2 =? 5^2
9 + 16 =? 25
25 = 25
Yes, that one is a right triangle.

5^2 + 12^2 =? 13^2
25 + 144 =? 169
169 = 168
Yes, that one is also a right triangle.

And all without 50 minutes of hand multiplication with the obligatory and confusing one little mistake that requires you to do it all again and even more slowly!

Answer 2

First thing, divide out the common factors. The first group has 11 common and the second group 10. You get
3,5,4 and 12,13,5. These are easier to compute the squares for. 3^2 +4^2 = 9 + 16 = 25 = 5^2
The second one 5^2 +12^2 = 25 + 144 = 169 =13^2

These two groups are examples of Pythagorean triples. Integers that make up the sides of right triangles.

Answer 3

In an isosceles right triangle the two equal sides must be the legs, since the hypotenuse is always the longest side in a right triangle. Let x be the length of a leg. Then, by Pythagoras, the length of the hypotenuse is: c = √(x² + x²) = √(2x²) = (√2) x But, you're told that c is 4 cm. longer than x: c = x + 4 (√2) x = x + 4 ... substituted for c (√2)x - x = 4 ... add -x to both sides (√2 - 1)x = 4 .... factor on the left x = 4/(√2 - 1) .... divide both sides by (√2 - 1) So, the answer is 4/(√2 - 1) centimeters.

Answer 4

In a right triangle, the Pythagorean theorem is true, so a^2 + b^2 = c^2. The hypotenuse is always the longest one.

33^2 + 44^2 =
1089 + 1936 =
3025 = 55^2

Answer 5

square each of the numbers of the triplet, add the smaller two and see if you get the largest.

eg. for 3, 4, 5 the squares are 9, 16, and 25 and since 9+16 = 25 the triangle is right angled.

for bigger numbers such as what you have, factor them:
eg the first one is 11 * 3, 11*5, and 11*4
now see if 3, 5, and 4 satisfy the rule, and if so the triangle is right angled.

Answer 6

a^2+b^2 must = c^2 for a right triangle
so 33^2+44^2=55^2, if this is a true statement then it is a right triangle

second
120^2+50^2=130^2 if we have a true statement then we have a right triangle.

Answer 7

Use the pythagorean theorem.
You could make the numbers "smaller" by dividing.
33,55,44 is the same as 3,5,4
5^2 = 3^2 + 4^2
120,130,50 is the same as 12, 13, 5
13^2 = 12^2 + 5^2

Answer 8

Hint: the Pythagorean Theorem works if and ONLY if the triangle is a right triangle.