Right triangle ABC has sides with length x, y and z. If triangle ABC

Let z = length of hypotenuse
Let x and y = lengths of the sides that make up the triangle’s right angle
NOTE: since triangle area = (base)(height)/2, we know that the area of triangle ABC = [color=red]xy/2

Given: x² + y² + z² = 98
The Pythagorean Theorem tells us that x² + y² = z²
So, we can write: x² + y² + (x² + y²) = 98
Rewrite: 2(x² + y²) = 98
Or we can write: x² + y² = 49

Since we already know that x² + y² = z², we can conclude that z² = 49, which means z = 7

If the perimeter = 17, we can write: x + y + z = 17
Since z = 7, we can write: x + y + 7 = 17
Simplify to get: x + y = 10

IMPORTANT: If we square both sides of the above equation, some nice things happen.
(x + y) ² = 10²
Expand: x² + 2xy + y² = 100
Rearrange to get: (x² + y²) + 2xy = 100
Replace x² + y² with 49 to get: 49 + 2xy = 100
Simplify: 2xy = 51

NOTE: Our goal is to determine the value of xy/2
So, take 2xy = 51 and divide both sides by 4 to get: xy/2 = 51/4 = 12.75

Answer:

Sure!!!

Please follow below attached fig.

As x^2+y^2=z^2----(1)

Also given x^2+y^2+z^2=98
putting value z^2 from (1)

2(x^2+y^2)= 98
or x^2+y^2=49------(2)

again putting x^2+y^2 =z^2 from (1)
z^2 =49 or z=7

Also given that perimeter x+y+z= 17
thus x+y= 17-7 =10---------(3)
sq both side to get
x^2+y^2+2xy =100
from (2)
2xy =100-49 =51
Also area = xy/2= 51/4 = 12.75

Ans 12.75

Ans A

please share the solution

Interesting question - how would you know to square both sides? Is there a rule?

\(x^2 + y^2 + z^2 = 98\)
\(z^2 = x^2 + y^2\)
\(2*z^2 = 98\)
z = 7

given perimeter: x + y + z = 17
if z = 7, than x+y = 10

\((x+y)^2 = x^2 + 2*x*y + y^2\)
\(x^2 + 2*x*y + y^2 = (x+y)^2 = 10^2 = 100\)
\(x^2 + y^2 = 49\)
\(2*x*y = 51\)
\(x*y = 25.5\)

Area of triangle = 1/2*x*y = 12.75

since, there are no specific mentions of the placement of angle A,B& C and the sides... is it always ok to consider that "B" will be 90deg and "z" will be hypotenuse ?

dear mdsaddamforgmat

the answer to your question is NO.

here as we calculated z=7 ,it means it is the longest side of rt. triangle ie hypotenuse.

try making other side the longest one,...u will not get a rt triangle then.

hope it helps.

Hi Members,

After I get the hypotenuse = 7 (diameter of the circle). When I compute the area as 1/2*7*3.5 (radius) = 12.25

I don't get the answer as 12.75. Can you help me understand where I am going wrong?

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