Let's begin by considering a $4 5^{\circ}$ - $4 5^{\circ}$ - $9 0^{\circ}$ triangle. Notice that this is an isosceles right triangle.

Let's apply the Pythagorean Theorem to the triangle above.

c^{2} = ℓ^{2} + ℓ^{2} \Rightarrow c = 2 ℓ^{2}

By simplifying the latter equation, we get that

c = 2 ℓ .

Thus, we are ready to write the relationship among the side lengths of a

4 5^{\circ}

-

4 5^{\circ}

-

9 0^{\circ}

triangle.

The legs have the same measure, and the length of the hypotenuse is $2$ times the length of each leg.

On the other hand, let's consider a $3 0^{\circ}$ - $6 0^{\circ}$ - $9 0^{\circ}$ triangle.

Next, let's consider a point $M$ such that it lies on $K L$ and $L M = a .$ With this new point, we will construct a second right triangle $△ J M L .$

Notice that $△ J K L ≅ △ J M L$ by the Side-Angle-Side (SAS) Congruence Theorem. Therefore, $m ∠ M = m ∠ K$ and $m ∠ M J L = m ∠ K J L .$

By the Angle Addition Postulate, we have that $m ∠ M J K = 3 0^{\circ} + 3 0^{\circ},$ so $m ∠ M J K = 6 0^{\circ} .$ In consequence, the three angles of $△ J K M$ are congruent.

As we can see, the triangle above is equiangular and so, by the Corollary to the Converse of the Base Angles Theorem , it is equilateral. This implies that $J K = 2 a .$ Now, let's turn back to the initial triangle.

Let's apply the Pythagorean Theorem to

△ J K L .

a^{2} + J L^{2} = (2 a)^{2}

Let's solve the equation above for

J L,

which is the length of the longer leg.

a^{2} + J L^{2} = (2 a)^{2}

CalcPow

Calculate power

a^{2} + J L^{2} = 4 a^{2}

▼

Solve for

J L

SubEqn

$LHS - a^{2} = RHS - a^{2}$

J L^{2} = 3 a^{2}

SqrtEqn

$LHS = RHS$

J L^{2} = 3 a^{2}

SqrtProd

$a \cdot b = a \cdot b$

J L^{2} = 3 a^{2}

SqrtToAbs

$a^{2} = ∣ a ∣$

∣ J L ∣ = 3 ∣ a ∣

Since

J L

and

a

are measures of sides of

△ J K L,

then both are positive and so we can remove the absolute value bars and get

J L = 3 a .

From the latter diagram, we can see that the hypotenuse is twice as long as the shorter leg, and the longer leg is $3$ times as long as the shorter leg. Finally, we are ready to write the relationship among the side lengths of a $3 0^{\circ}$ - $6 0^{\circ}$ - $9 0^{\circ}$ triangle.

The length of the longer leg is $3$ times the length of the shorter leg, and the length of the hypotenuse is twice the length of the shorter leg.

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