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# Pythagorean theorem II

More Pythagorean Theorem examples. Introduction to 45-45-90 triangles. Created by Sal Khan.

## Want to join the conversation?

• At and on all the exercise problems, why do they claim the shorter leg to be longer than the oter one when it is actually smaller? That really confused me.
• See, it does not matter which leg is smaller or bigger.
Only the hypotenuse matters -

Example - A triangle with sides of measurement 3,4 & 5 than you can know that the hypotenuse is the longest leg.

But if we had to find hypotenuse than we can add in any way but in subtraction we have to subtract the smaller length from bigger!
• how do you get a 5 square root of 2??
• 50's factors are: 2*25, but 25 factors out more so 2*5*5, then as 2 of the 5's in the square root sign are same, then take it out of the square root and you are left with 5 root 2~ :)
• I don't understand the part when he said "The part that they don't share are equivalent" I need help on this.
• Each angle touches two sides of the triangle. If two angles are equal, the two sides other than the side that both those angles touch, are equal. If you still don't understand, you should watch the video again.
• I hope this is the right place to post this question:

How do I find the legs of a triangle using only the hypotenuse and the area?
• This is a fairly easy, but very tedious problem. Here is how to do it:
Use simultaneous solution from the area formula and the Pythagorean theorem
a² + b² = c²
½ ab = A (area)
Thus,
ab = 2A
b = 2A/a
Thus,
b² = [2A/a]²
b² = 4A²/a²
Substituting into Pythagorean Theorem:
a² + b² = c²
a² +4A²/a² = c²
a⁴ + 4A² = c²a²
a⁴ - c²a² + 4A² = 0
Let x = a²
x² - c²x + 4A² = 0
x =½ [c² ±√(c⁴-16A²)]
a² = ½ [c² ±√(c⁴-16A²)]
This will give you two answers. One of these will, of course, be a². The other will be b². Just take the positive square roots of those two answers to get the two legs' lengths.
Example. A right triangle has a hypotenuse of 5 cm and an area of 6 cm². What are the lengths of the legs?
a² = ½ [c² ±√(c⁴-16A²)]
a² = ½ [5² ±√(5⁴-16(6)²)]
a² = ½ [25 ±√(625-16(36))]
a² = ½ [25 ±√(625-576)]
a² = ½ [25 ±√(49)]
a² = ½ [25 ± 7]
a² = ½ [18] OR a² = ½ (32)
a² = 9 OR a² = 16
a = 3 OR a = 4
Thus the two leg's lengths are 3 cm and 4 cm.
• if a squared+ b squared = c squared wouldn't a+b=c?
• When Sal said that the lengths of a and b are both equal to 5 because the angles of 45 degrees in the triangle are the same, isn't that true for all right triangles? Because right triangles always have 90degree angle with 45degree angle and another 45degree angle right? And if that was true, why aren't all of the right triangles sides a and b the same?
(1 vote)
• No, because if you have a longer and shorter leg, you can have a 30-60-90 triangle, which is still a right triangle.
• How can the square root of 50 turned to the squared root of 2 times 5
• √50
50=25*2.
√(25*2)
√25*√2
√25=5.
5√2
I hope this helps!
(1 vote)
• A 90 degree angle always has two 45 degree angles and a 90 degree angle. You said that if 2 angles are equal in a triangles then the sides they are not sharing with each other are equal as well. But in my math textbook it says that sides of right angled triangle are always in ratio 3:4:5.
(1 vote)
• The sides of a right triangle are not always in the ratio 3 : 4 : 5. That is just one example of a primitive Pythagorean triple. There are an infinite number of possible ratios for right triangles and 3 : 4 : 5 is just one possibility.