120 lines
3.3 KiB
Markdown
120 lines
3.3 KiB
Markdown
---
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id: 5900f3c71000cf542c50feda
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title: 'Problem 91: Right triangles with integer coordinates'
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challengeType: 5
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forumTopicId: 302208
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dashedName: problem-91-right-triangles-with-integer-coordinates
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---
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# --description--
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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 ${\Delta}OPQ$.
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<img class="img-responsive center-block" alt="a graph plotting points P (x_1, y_1) and Q(x_2, y_2) at integer coordinates that are joined to the origin O (0, 0)" src="https://cdn-media-1.freecodecamp.org/project-euler/right-triangles-integer-coordinates-1.png" style="background-color: white; padding: 10px;" />
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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, $0 ≤ x_1, y_1, x_2, y_2 ≤ 2$.
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<img class="img-responsive center-block" alt="a diagram showing the 14 triangles containing a right angle that can be formed when each coordinate is between 0 and 2" src="https://cdn-media-1.freecodecamp.org/project-euler/right-triangles-integer-coordinates-2.png" style="background-color: white; padding: 10px;" />
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Given that $0 ≤ x_1, y_1, x_2, y_2 ≤ limit$, how many right triangles can be formed?
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# --hints--
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`rightTrianglesIntCoords(2)` should return a number.
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```js
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assert(typeof rightTrianglesIntCoords(2) === 'number');
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```
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`rightTrianglesIntCoords(2)` should return `14`.
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```js
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assert.strictEqual(rightTrianglesIntCoords(2), 14);
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```
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`rightTrianglesIntCoords(10)` should return `448`.
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```js
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assert.strictEqual(rightTrianglesIntCoords(10), 448);
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```
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`rightTrianglesIntCoords(25)` should return `3207`.
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```js
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assert.strictEqual(rightTrianglesIntCoords(25), 3207);
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```
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`rightTrianglesIntCoords(50)` should return `14234`.
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```js
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assert.strictEqual(rightTrianglesIntCoords(50), 14234);
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```
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# --seed--
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## --seed-contents--
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```js
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function rightTrianglesIntCoords(limit) {
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return true;
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}
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rightTrianglesIntCoords(2);
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```
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# --solutions--
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```js
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function rightTrianglesIntCoords(limit) {
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function isRightTriangle(points) {
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for (let i = 0; i < points.length; i++) {
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const pointA = points[i];
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const pointB = points[(i + 1) % 3];
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const pointC = points[(i + 2) % 3];
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const vectorAB = [pointB[0] - pointA[0], pointB[1] - pointA[1]];
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const vectorAC = [pointC[0] - pointA[0], pointC[1] - pointA[1]];
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if (isRightAngleBetween(vectorAB, vectorAC)) {
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return true;
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}
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}
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return false;
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}
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function isRightAngleBetween(vector1, vector2) {
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return vector1[0] * vector2[0] + vector1[1] * vector2[1] === 0;
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}
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function getSetKey(points) {
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return (
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'0.0,' +
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points
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.sort((a, b) => a[0] - b[0])
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.map(point => point.join('.'))
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.join(',')
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);
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}
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const pointO = [0, 0];
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const rightTriangles = new Set();
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for (let x1 = 1; x1 <= limit; x1++) {
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for (let y1 = 0; y1 <= limit; y1++) {
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const pointP = [x1, y1];
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for (let x2 = 0; x2 <= limit; x2++) {
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for (let y2 = 1; y2 <= limit; y2++) {
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const pointQ = [x2, y2];
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if (pointP[0] === pointQ[0] && pointP[1] === pointQ[1]) {
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continue;
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}
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if (isRightTriangle([pointO, pointP, pointQ])) {
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rightTriangles.add(getSetKey([pointP, pointQ]));
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}
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}
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}
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}
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}
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return rightTriangles.size;
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}
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```
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