A right triangle has perimeter 20. If the sum of the squares of the triangle's three sides is 162,
what is the area of the triangle?
Let the three sides are a, b and c......think a and b are sides and c as the hypotenuse of the right angle traingle.
a+b+c = 20 , a^2+b^2+c^2 = 162
from phythogoreous theorem , a^2+b^2 = c^2
use below formula
(a+b+c) ^2 = a^2+b^2+c^2+2ab+2bc+2ca
20 ^ 2 = 162 + 2ab+2bc+2ca
ab+bc+ca = 400 - 162/2 = 119
since a^2+b^2 = c^2 and a^2+b^2+c^2 = 162 => 2 * c^2 = 162 => c^2 = 81 => c = 9
ab+bc+ca = 119 => ab+9b+9a (substitute the value of c from above)
ab+9(a+b) = 119
ab+9(11) = 119 ( since, a+b+c = 20 and c= 9 => a+b =11)
=> ab = 119 -99 = 20
=> ab = 20
area of triangle = 1/2* a * b = 1/2 * 20 = 10
Therefor area of triangle is 10
Geo
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Rahul@gurome
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Say, length of sides of the triangle are a, b, and c and c is the hypotenuse. Thus, area of the triangle = (ab)/2yellowho wrote:A right triangle has perimeter 20. If the sum of the squares of the triangle's three sides is 162, what is the area of the triangle?
Hence, (a + b + c) = 20 and (a² + b² + c²) = 162
As, (a² + b²) = c²
=> (c² + c²) = 162
=> c² = 81
=> c = 9
Hence, (a + b) = 11
=> (a + b)² = (a² + 2ab + b²) = 121
As, (a² + b²) = c² = 81
=> (2ab + 81) = 121
=> 2ab = 40
=> (ab)/2 = 10
Hence area of the triangle is 10 sq. units.
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yellowho
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Wow. there u go. huge difference in time.
[quote="Rahul@gurome"][quote="yellowho"]A right triangle has perimeter 20. If the sum of the squares of the triangle's three sides is 162, what is the area of the triangle?[/quote]
Say, length of sides othe triangle are a, b, and c and c is the hypotenuse. Thus, area of the triangle = (ab)/2
Hence, (a + b + c) = 20 and (a² + b² + c²) = 162
As, (a² + b²) = c²
=> (c² + c²) = 162
=> c² = 81
=> c = 9
Hence, (a + b) = 11
=> (a + b)² = (a² + 2ab + b²) = 121
As, (a² + b²) = c² = 81
=> (2ab + 81) = 121
=> 2ab = 40
=> (ab)/2 = 10
Hence area of the triangle is 10 sq. units.[/quote]
[quote="Rahul@gurome"][quote="yellowho"]A right triangle has perimeter 20. If the sum of the squares of the triangle's three sides is 162, what is the area of the triangle?[/quote]
Say, length of sides othe triangle are a, b, and c and c is the hypotenuse. Thus, area of the triangle = (ab)/2
Hence, (a + b + c) = 20 and (a² + b² + c²) = 162
As, (a² + b²) = c²
=> (c² + c²) = 162
=> c² = 81
=> c = 9
Hence, (a + b) = 11
=> (a + b)² = (a² + 2ab + b²) = 121
As, (a² + b²) = c² = 81
=> (2ab + 81) = 121
=> 2ab = 40
=> (ab)/2 = 10
Hence area of the triangle is 10 sq. units.[/quote]
