freeCodeCamp/curriculum/challenges/italian/10-coding-interview-prep/project-euler/problem-91-right-triangles-with-integer-coordinates.md

id	title	challengeType	forumTopicId	dashedName
5900f3c71000cf542c50feda	Problem 91: Right triangles with integer coordinates	5	302208	problem-91-right-triangles-with-integer-coordinates

--description--

The points {P}(x_1, y_1) and {Q}(x_2, y_2) are plotted at integer co-ordinates and are joined to the origin, {O}(0, 0), to form {\Delta}OPQ.

There are exactly fourteen triangles containing a right angle that can be formed when each co-ordinate lies between 0 and 2 inclusive; that is, 0 ≤ x_1, y_1, x_2, y_2 ≤ 2.

Given that 0 ≤ x_1, y_1, x_2, y_2 ≤ limit, how many right triangles can be formed?

--hints--

rightTrianglesIntCoords(2) should return a number.

assert(typeof rightTrianglesIntCoords(2) === 'number');

rightTrianglesIntCoords(2) should return 14.

assert.strictEqual(rightTrianglesIntCoords(2), 14);

rightTrianglesIntCoords(10) should return 448.

assert.strictEqual(rightTrianglesIntCoords(10), 448);

rightTrianglesIntCoords(25) should return 3207.

assert.strictEqual(rightTrianglesIntCoords(25), 3207);

rightTrianglesIntCoords(50) should return 14234.

assert.strictEqual(rightTrianglesIntCoords(50), 14234);

--seed--

--seed-contents--

function rightTrianglesIntCoords(limit) {

  return true;
}

rightTrianglesIntCoords(2);

--solutions--

function rightTrianglesIntCoords(limit) {
  function isRightTriangle(points) {
    for (let i = 0; i < points.length; i++) {
      const pointA = points[i];
      const pointB = points[(i + 1) % 3];
      const pointC = points[(i + 2) % 3];
      const vectorAB = [pointB[0] - pointA[0], pointB[1] - pointA[1]];
      const vectorAC = [pointC[0] - pointA[0], pointC[1] - pointA[1]];

      if (isRightAngleBetween(vectorAB, vectorAC)) {
        return true;
      }
    }
    return false;
  }

  function isRightAngleBetween(vector1, vector2) {
    return vector1[0] * vector2[0] + vector1[1] * vector2[1] === 0;
  }

  function getSetKey(points) {
    return (
      '0.0,' +
      points
        .sort((a, b) => a[0] - b[0])
        .map(point => point.join('.'))
        .join(',')
    );
  }

  const pointO = [0, 0];
  const rightTriangles = new Set();
  for (let x1 = 1; x1 <= limit; x1++) {
    for (let y1 = 0; y1 <= limit; y1++) {
      const pointP = [x1, y1];
      for (let x2 = 0; x2 <= limit; x2++) {
        for (let y2 = 1; y2 <= limit; y2++) {
          const pointQ = [x2, y2];
          if (pointP[0] === pointQ[0] && pointP[1] === pointQ[1]) {
            continue;
          }
          if (isRightTriangle([pointO, pointP, pointQ])) {
            rightTriangles.add(getSetKey([pointP, pointQ]));
          }
        }
      }
    }
  }
  return rightTriangles.size;
}