randRange( 2, 7 )

In the right triangle shown, AC = BC = AC. What is AB?

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

This is a 45-45-90 triangle because the two legs are congruent.

Thus, the hypotenuse is \sqrt{2} times as long as each of the legs.

AB = 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", "?", "?", AB );
AB * AB / 2

This is a 45-45-90 triangle because the two legs are congruent.

Thus, each leg is \frac{1}{\sqrt{2}} times as long as the hypotenuse.

AC = BC = \frac{1}{\sqrt{2}} AB = 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", "?", "?", AB + "\\sqrt{2}" );
AB * AB

This is a 45-45-90 triangle because the two legs are congruent.

Thus, each leg is \frac{1}{\sqrt{2}} times as long as the hypotenuse.

AC = BC = \frac{1}{\sqrt{2}} AB \sqrt{2} = AB

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

In the right triangle shown, mAB and BC = BC + BCrs. How long is AB?

betterTriangle( 1, sqrt(3), "A", "B", "C", BC + BCrs, "", "?" );
4 * BC * BC * BCr

This is a 30-60-90 triangle.

Thus, the hypotenuse is twice as long as the shorter leg.

AB = 2 \cdot BC = 2 \cdot (BC + BCrs) = ( 2 * 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] ]) randFromArray([ "\\angle A = 30^\\circ", "\\angle B = 60^\\circ" ])

In the right triangle shown, mAB and AC = AC + ACrs. How long is AB?

betterTriangle( 1, sqrt(3), "A", "B", "C", "", AC + ACrs, "?" );
AB

This is a 30-60-90 triangle.

Thus, the hypotenuse is \frac{2}{\sqrt{3}} times as long as the longer leg.

AB = \frac{2}{\sqrt{3}} \cdot AC = \frac{2}{\sqrt{3}} \cdot (AC + ACrs) = ABs

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

In the right triangle shown, mAB and AB = ( 2 * BC ) + BCrs. How long is BC?

betterTriangle( 1, sqrt(3), "A", "B", "C", "?", "", ( 2 * BC ) + BCrs );
BC * BC * BCr

This is a 30-60-90 triangle.

Thus, the shorter leg is half as long as the hypotenuse.

BC = \frac12 \cdot AB = \frac12 \cdot (( 2 * BC ) + BCrs) = ( 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] ]) randFromArray([ "\\angle A = 30^\\circ", "\\angle B = 60^\\circ" ])

In the right triangle shown, mAB and AB = ABs. How long is AC?

betterTriangle( 1, sqrt(3), "A", "B", "C", "", "?", ABs );
AC * AC * ACr

This is a 30-60-90 triangle.

Thus, the longer leg is \frac{\sqrt{3}}{2} times as long as the hypotenuse.

AC = \frac{\sqrt{3}}{2} \cdot AB = \frac{\sqrt{3}}{2} \cdot (ABs) = AC + ACrs