randRange(2, 7)

In the right triangle shown, `AC = BC = AC`.

What is `AB`?

betterTriangle( 1, 1, "A", "B", "C", AC, AC, "x" );
AC * AC * 2

We know the length of each leg, and want to find the length of the hypotenuse. What mathematical relationship is there between a right triangle's leg and its hypotenuse?

`\qquad AC^2 + BC^2 = x^2`

`\qquad AC^2 + AC^2 = x^2`

`\qquad x^2 = 2 \cdot AC^2`

`\qquad x = \sqrt{2 \cdot AC^2}`

`\qquad x = \sqrt{2} \cdot \sqrt{AC^2}`

`\qquad x = AC\sqrt{2}`

2 * randRange(2, 6)

In the right triangle shown, `AC = BC` and `AB = AB`.

How long are each of the legs?

betterTriangle( 1, 1, "A", "B", "C", "x", "x", AB );
AB * AB / 2

We know the length of the hypotenuse, and want to find the length of each leg. What mathematical relationship is there between a right triangle's legs and its hypotenuse?

`\qquad x^2 + x^2 = AB^2`

`\qquad 2 \cdot x^2 = AB^2`

`\qquad x^2 = AB * AB / 2`

`\qquad x = \sqrt{AB * AB / 2}`

`\qquad x = \sqrt{AB * AB / 4 \cdot 2}`

`\qquad x = \sqrt{AB * AB / 4} \cdot \sqrt{2}`

`\qquad x = AB/2 \sqrt{2}`

2 * randRange(2, 6)

In the right triangle shown, `AC = BC` and `AB = AB\sqrt{2}`.

How long are each of the legs?

betterTriangle( 1, 1, "A", "B", "C", "x", "x", AB + "\\sqrt{2}" );
AB * AB

We know the length of the hypotenuse, and want to find the length of each leg. What mathematical relationship is there between a right triangle's legs and its hypotenuse?

`\qquad x^2 + x^2 = AB^2`

`\qquad 2 \cdot x^2 = (AB\sqrt{2})^2`

`\qquad 2 \cdot x^2 = AB^2 \cdot (\sqrt{2})^2`

`\qquad 2 \cdot x^2 = AB * AB \cdot 2`

`\qquad x^2 = AB * AB`

`\qquad x = AB`

randRange(2, 6) randFromArray([[1, ""], [3, "\\sqrt{3}"]]) BC + BCrs rand(2) ["\\angle A = 30^\\circ", "\\angle B = 60^\\circ"][ANGLE]

In the right triangle shown, `mAB` and `BC = BC + BCrs`.

How long is `AB`?

betterTriangle(1, sqrt(3), "A", "B", "C", BC + BCrs, "", "x"); if (ANGLE === 0) { arc([0, 5 * sqrt(3) / 2], 0.8, 270, 300); label([-0.1, (5 * sqrt(3) / 2) - 1], "{30}^{\\circ}", "below right"); } else { arc([5/2,0], 0.5, 120, 180); label([5/2-0.2, 0], "{60}^{\\circ}", "above left"); }
4 * BC * BC * BCr

We know the length of a leg, and want to find the length of the hypotenuse. What is the relationship between the sides of a `30 - 60 - 90` triangle?

This is a `30-60-90` triangle with a hypotenuse of length `1`.

betterTriangle(1, sqrt(3), "A", "B", "C", "\\dfrac{1}{2}", "\\dfrac{\\sqrt{3}}{2}", 1); arc([0, 5 * sqrt(3) / 2], 0.8, 270, 300); arc([5/2,0], 0.5, 120, 180); label([-0.1, (5 * sqrt(3) / 2) - 1], "{30}^{\\circ}", "below right"); label([5/2-0.2, 0], "{60}^{\\circ}", "above left");

The ratio of `AB : BC` is `1 : \dfrac{1}{2}`.

Therefore, `\dfrac{x}{BC + BCrs} = \dfrac{1}{\frac{1}{2}} = 2`.

`x = 2 \cdot BC + BCrs`

`x = BC * 2 + BCrs`

3 * randRange( 2, 6 ) randFromArray([ [1, "", (AC * 2 / 3) + "\\sqrt{3}", AC * AC * 4 / 3], [3, "\\sqrt{3}", (AC * 2), AC * AC * 4] ]) AC + ACrs rand(2) ["\\angle A = 30^\\circ", "\\angle B = 60^\\circ"][ANGLE]

In the right triangle shown, `mAB` and `AC = AC + ACrs`.

How long is `AB`?

betterTriangle(1, sqrt(3), "A", "B", "C", "", AC + ACrs, "x"); if (ANGLE === 0) { arc([0, 5 * sqrt(3) / 2], 0.8, 270, 300); label([-0.1, (5 * sqrt(3) / 2) - 1], "{30}^{\\circ}", "below right"); } else { arc([5/2,0], 0.5, 120, 180); label([5/2-0.2, 0], "{60}^{\\circ}", "above left"); }
AB

We know the length of a leg, and want to find the length of the hypotenuse. What is the relationship between the sides of a `30 - 60 - 90` triangle?

This is a `30-60-90` triangle with a hypotenuse of length `1`.

betterTriangle(1, sqrt(3), "A", "B", "C", "\\dfrac{1}{2}", "\\dfrac{\\sqrt{3}}{2}", 1); arc([0, 5 * sqrt(3) / 2], 0.8, 270, 300); arc([5 / 2,0], 0.5, 120, 180); label([-0.1, (5 * sqrt(3) / 2) - 1], "{30}^{\\circ}", "below right"); label([5 / 2 - 0.2, 0], "{60}^{\\circ}", "above left");

The ratio of `AB : AC` is `1 : \dfrac{\sqrt{3}}{2}`.

Therefore, `\dfrac{x}{AC + ACrs} = \dfrac{1}{\frac{\sqrt{3}}{2}} = \dfrac{2}{\sqrt{3}}`.

`x = \dfrac{2}{\sqrt{3}} \cdot AC + ACrs`

`x = \dfrac{2 * AC}{\sqrt{3}}`

`x = \dfrac{2 * AC}{\sqrt{3}} \cdot \dfrac{\sqrt{3}}{\sqrt{3}}`

`x = \dfrac{2 * AC \cdot \sqrt{3}}{3}`

So, `x = ABs`.

randRange( 2, 6 ) randFromArray([ [1, ""], [3, "\\sqrt{3}"] ]) 2*BC + BCrs rand(2) ["\\angle A = 30^\\circ", "\\angle B = 60^\\circ"][ANGLE]

In the right triangle shown, `mAB` and `AB = 2 * BC + BCrs`.

How long is `BC`?

betterTriangle(1, sqrt(3), "A", "B", "C", "x", "", 2 * BC + BCrs); if (ANGLE === 0) { arc([0, 5 * sqrt(3) / 2], 0.8, 270, 300); label([-0.1, (5 * sqrt(3) / 2) - 1], "{30}^{\\circ}", "below right"); } else { arc([5/2,0], 0.5, 120, 180); label([5/2-0.2, 0], "{60}^{\\circ}", "above left"); }
BC * BC * BCr

We know the length of the hypotenuse, and want to find the length of the shortest side. What is the relationship between the sides of a `30 - 60 - 90` triangle?

This is a `30-60-90` triangle with a hypotenuse of length `1`.

betterTriangle(1, sqrt(3), "A", "B", "C", "\\dfrac{1}{2}", "\\dfrac{\\sqrt{3}}{2}", 1); arc([0, 5 * sqrt(3) / 2], 0.8, 270, 300); arc([5/2,0], 0.5, 120, 180); label([5/2-0.2, 0], "{60}^{\\circ}", "above left"); label([-0.1, (5 * sqrt(3) / 2) - 1], "{30}^{\\circ}", "below right");

The ratio of `BC : AB` is `\dfrac{1}{2} : 1`.

Therefore, `\dfrac{x}{ABdisp} = \dfrac{1}{2}`.

`x = BC + BCrs`

3 * randRange( 2, 6 ) randFromArray([ [1, "", (AC * 2 / 3) + "\\sqrt{3}", AC * AC * 4 / 3], [3, "\\sqrt{3}", (AC * 2), AC * AC * 4] ]) rand(2) ["\\angle A = 30^\\circ", "\\angle B = 60^\\circ"][ANGLE]

In the right triangle shown, `mAB` and `AB = ABs`.

How long is `AC`?

betterTriangle(1, sqrt(3), "A", "B", "C", "", "x", ABs); if (ANGLE === 0) { arc([0, 5 * sqrt(3) / 2], 0.8, 270, 300); label([-0.1, (5 * sqrt(3) / 2) - 1], "{30}^{\\circ}", "below right"); } else { arc([5/2,0], 0.5, 120, 180); label([5/2-0.2, 0], "{60}^{\\circ}", "above left"); }
AC * AC * ACr

We know the length of the hypotenuse, and want to find the length of the longest side. What is the relationship between the sides of a `30 - 60 - 90` triangle?

This is a `30-60-90` triangle with a hypotenuse of length `1`.

betterTriangle(1, sqrt(3), "A", "B", "C", "\\dfrac{1}{2}", "\\dfrac{\\sqrt{3}}{2}", 1); arc([0, 5 * sqrt(3) / 2], 0.8, 270, 300); arc([5 / 2,0], 0.5, 120, 180); label([-0.1, (5 * sqrt(3) / 2) - 1], "{30}^{\\circ}", "below right"); label([5 / 2 - 0.2, 0], "{60}^{\\circ}", "above left");

The ratio of `AC : AB` is `\dfrac{\sqrt{3}}{2} : 1`.

Therefore, `\dfrac{x}{ABs} = \dfrac{\sqrt{3}}{2}`.

`x = \dfrac{\sqrt{3}}{2} \cdot ABs`

`x = AC / 3 \cdot \sqrt{3} \cdot \sqrt{3}`

So, `x = AC + ACrs`.