Answer to Question #100780 in Trigonometry for Sidan

Question

Find all heronian triangles with one altitude 40 cm. (I just want the sides like 401,401,798)

Answer 1

Using the Heron's formula will determine the area and height for each triangle

$\quad A = \sqrt{s(s-a)(s-b)(s-c)} \quad \text{, where } s = \cfrac{a+b+c}2 \\ \\ \quad h_b = \cfrac{2A}b$

(41, 18, 41) A = 360, h = 40

(41, 39, 50) A = 780, h = 40

(41, 51, 58) A = 1020, h = 40

(41, 84, 85) A = 1680, h = 40

(41, 105, 104) A = 2100, h = 40

(41, 207, 202) A = 4140, h = 40

(41, 408, 401) A = 8160, h = 40

(50, 60, 50) A = 1200, h = 40

(50, 72, 58) A = 1440, h = 40

(50, 105, 85) A = 2100, h = 40

(50, 126, 104) A = 2520, h = 40

(50, 228, 202) A = 4560, h = 40

(50, 429, 401) A = 8580, h = 40

(58, 84, 58) A = 1680, h = 40

(58, 117, 85) A = 2340, h = 40

(58, 138, 104) A = 2760, h = 40

(58, 240, 202) A = 4800, h = 40

(58, 441, 401) A = 8820, h = 40

(85, 150, 85) A = 3000, h = 40

(85, 171, 104) A = 3420, h = 40

(85, 273, 202) A = 5460, h = 40

(85, 474, 401) A = 9480, h = 40

(104, 192, 104) A = 3840, h = 40

(104, 294, 202) A = 5880, h = 40

(104, 495, 401) A = 9900, h = 40

(202, 396, 202) A = 7920, h = 40

(202, 597, 401) A = 11940, h = 40

(401, 798, 401) A = 15960, h = 40

Comment: First I find all right triangles with integer side lengths having one leg of length 40. Then combine it with each other.

<code js>

{

const a=[];

// Search for right-angled triangles with integer side lengths having one leg of length 40

for(let c, b=1; b<1e4; b++) {

c=Math.sqrt(1600 + b*b); // hypotenuse

if(Number.isInteger(c)) a.push([b,c])

}

// Combine finded triangles

for(let i=0, L=a.length; i<L; i++)

for(let j=i; j<L; j++)

console.log('%d, %d, %d',

a[i][1], // hypotenuse of one triangle

a[i][0]+a[j][0], // Sum of legs

a[j][1] // hypotenuse of other triangle

);

}

</code>