Question

the length of one side of △ptv is given. use the relationship between the sides of a 30°-60°-90° triangle to find the lengths of the other two sides. if your answer is $5\sqrt{2}$, please type \5sqrt2\ - no space. given $vt = 5\sqrt{3}$. complete the table and find the missing sides (radical form)

Explanation:

Step1: Match side to 30-60-90 rule

In a 30-60-90 triangle, the side opposite $60^\circ$ is $x\sqrt{3}$. Here, $VT = 5\sqrt{3}$ is opposite $\angle P = 60^\circ$, so:
$x\sqrt{3} = 5\sqrt{3}$
Solve for $x$: $x = 5$

Step2: Find side opposite 30°

The side opposite $30^\circ$ is $x$. This side is $PT$, so:
$PT = x = 5$

Step3: Find hypotenuse (opposite 90°)

The hypotenuse (opposite $90^\circ$) is $2x$. This side is $PV$, so:
$PV = 2x = 2\times5 = 10$

Step4: Fill the table

• $30^\circ$ side: $x = 5$
• $60^\circ$ side: $x\sqrt{3} = 5\sqrt{3}$
• $90^\circ$ side: $2x = 10$

Answer:

Table:

$30^\circ$	$60^\circ$	$90^\circ$

Missing sides:

$PT = 5$
$PV = 10$