Question

apply the 45°-45°-90° triangle theorem to find the length of the hypotenuse of the triangle.
if the length of each leg of the triangle is $15\sqrt{2}$ in, what is the length of the hypotenuse(c)?
$15\sqrt{2}\sqrt{15}$ in
30 in
$2 \times 15\sqrt{2}$ in
$\sqrt{15\sqrt{2}}$ in

1. choose the correct answer.

apply the 30°-60°-90° triangle theorem to find the length of the longer leg of a triangle if the length of the shorter leg is $5\sqrt{3}$ inches.
$\sqrt{2} \times 5\sqrt{3}$ inches
15 inches
$3 \times 5\sqrt{3}$ inches
$2 \times 5\sqrt{3}$ inches

Explanation:

First Sub - Question (45° - 45° - 90° Triangle)

Step1: Recall the 45 - 45 - 90 theorem

In a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle, the length of the hypotenuse \(c\) is given by \(c = l\times\sqrt{2}\), where \(l\) is the length of each leg.
Here, \(l = 15\sqrt{2}\) in.

Step2: Calculate the hypotenuse

Substitute \(l = 15\sqrt{2}\) into the formula:
\(c=(15\sqrt{2})\times\sqrt{2}\)
We know that \(\sqrt{a}\times\sqrt{a}=a\), so \(\sqrt{2}\times\sqrt{2} = 2\).
Then \(c = 15\times2=30\) in.

Second Sub - Question (30° - 60° - 90° Triangle)

Step1: Recall the 30 - 60 - 90 theorem

In a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle, the length of the longer leg \(L\) is given by \(L = s\times\sqrt{3}\), where \(s\) is the length of the shorter leg.
Here, \(s = 5\sqrt{3}\) inches.

Step2: Calculate the longer leg

Substitute \(s = 5\sqrt{3}\) into the formula:
\(L=(5\sqrt{3})\times\sqrt{3}\)
Since \(\sqrt{3}\times\sqrt{3}=3\), we have \(L = 5\times3 = 15\) inches.

Answer:

First Sub - Question: 30 in
Second Sub - Question: 15 inches