The perimeter of a right angled triangle is 60cm, it's hypotenuse is 26cm find its area
Answers
Answer:
Step-by-step explanation:
Let’s call each side a, b, and c respectively, where c is the hypotenuse.
If a+b+c=60 and a2+b2=c2 and we know that c=26 , then we have enough information to work this out.
The area can be written as ab2 , which is the same as base * height over 2.
Therefore, using the above equations,
a+b+26=60
a+b=34
Also,
a2+b2=676
One way to solve this problem is to square the first equation,
(a+b)2=342
a2+b2+2ab=1156
Oh, cool! We can substitute the a2+b2 in the above equation with our second equation!
a2+b2=676
676+2ab=1156
Subtract 676 from both sides, we get:
2ab=480
Since the area of said right triangle can be expressed as ab/2 , all we have to do now is divide by 4 both sides of the equation!
ab2=120
The area of this triangle is 120cm2 . If you’re wondering what the sides are, they are 2 4 and 10 .
One way to get this is to check the factors of 26:
1,2,13 and 26 .
The only Pythagorean triple with relatively prime integers that include one of these numbers is the set (5,12,13) .
So multiplying every side by 2, we get the set (10,24,26) .
There’s another way… using the equations a+b=34 and ab=240 . If you’re interested in this solution, read on:
a=34−b
b(34−b)=240
Expanding, we get 34b−b2=240 . With the formula
x=−b±b2−4ac√2a
Where ax2+x+c=0 , we can express this as −b2+34b−240=0 .
a=−1
b=34
c=−240
x=−34±342−960√−2
x=−34±196√−2
x=−34±14−2
x1=10,x2=24
Now thanks to this handy formula, we now have the possible values of b !
(a,b)=(10,24) or (24,10)
Since it’s a triangle it doesn’t really matter. The side lengths are 10, 24 and 26.