what is the lengths of a right triangle if the hypotenuse is 12 and the area of the triangle 24?

The formula for area of a triangle is A=b*h/2, where b and h base and height of a triangle. In the case of a right triangle, they are the legs. So we have two unknowns and 2 equations.

 

24=b*h/2

b^2 + h^2 = 144   (Pythagorean theorem)

 

Solve the first equation for either b or h and substitute into the second.

 

24=b*h/2

48=b*h  (multiply both sides by 2)

b=48/h   (divide both sides by h)

 

(48/h)^2 + h^2 = 144 (substitute 48/h for b)

2304/h^2 + h^2 = 144

2304/h^2 + h^4/h^2 = 144  (find common denominator for left side)

2304 + h^4 = 144*h^2   (multiply both sides by h^2)

 

This is really a quadratic equation. You can make this clearer by letting x = h^2

 

2304 + x^2 = 144x

x^2 - 144x + 2304 = 0

 

Use the quadratic formula to solve.

 

x = (144 ± sqrt((-144)^2-4*1*2304)))/2 ≈ 125.665, 18.334

If x = h^2 then sqrt(x) = h

h ≈ 11.210 or 4.282

 

Then plug into b = 48/h to get b. You'll find that if you choose 11.210 for h you'll bet 4.282 for b and if you choose 4.282 for h you'll get 11.210 for b.

 

So the legs of the triangle are approximately 4.282 and 11.210 inches.

 

Thanks for the interesting problem.

Area of a triangle is (B * H)/2 

The area is 24   .---- B*H=48 ---- B=48/H

 

the hypotenuse is 12^2=B^2+H^2

Replace B by (48 / H)

(48 / H)^2 + H^2 = 144

Now you just have to solve it to find H.

When you find H, you will be able to solve for B with this equation B=48/H