Question

find the requested length (dont forget appropriate units)

1. the length of the diagonal of a square is 12 inches. find the length of one side of the square.
2. the length of one side of an equilateral triangle is ( 6sqrt{3} ) meters. find the length of the altitude of the triangle.
3. the length of the altitude of an equilateral triangle is 12 feet. find the length of one side of the equilateral triangle. what is the perimeter of the equilateral triangle?
4. the perimeter of an equilateral triangle is 39 cm. find the length of the altitude of the triangle.
5. the length of the diagonal of a square is 18 mm. find the perimeter of the square.
6. the diagonal of a rectangle is 12 in and intersect at an angle to make a ( 60^circ ) angle. find the perimeter of the rectangle.

Explanation:

Step1: Relate square diagonal to side

For a square, diagonal $d = s\sqrt{2}$, so $s = \frac{d}{\sqrt{2}}$.
Given $d=12$ in, $s = \frac{12}{\sqrt{2}} = 6\sqrt{2}$ in.

Step2: Use equilateral triangle altitude formula

Altitude $h = \frac{\sqrt{3}}{2}s$.
Given $s=6\sqrt{3}$ m, $h = \frac{\sqrt{3}}{2} \times 6\sqrt{3} = 9$ m.

Step3: Find side from equilateral altitude

Rearrange $h = \frac{\sqrt{3}}{2}s$ to $s = \frac{2h}{\sqrt{3}}$.
Given $h=12$ ft, $s = \frac{2 \times 12}{\sqrt{3}} = 8\sqrt{3}$ ft.
Perimeter $P=3s = 3 \times 8\sqrt{3} = 24\sqrt{3}$ ft.

Step4: Find side from equilateral perimeter

Perimeter $P=3s$, so $s = \frac{P}{3}$.
Given $P=39$ cm, $s = \frac{39}{3}=13$ cm.
Altitude $h = \frac{\sqrt{3}}{2}s = \frac{13\sqrt{3}}{2}$ cm.

Step5: Find square side from diagonal

$s = \frac{d}{\sqrt{2}}$.
Given $d=18$ mm, $s = \frac{18}{\sqrt{2}} = 9\sqrt{2}$ mm.
Perimeter $P=4s = 4 \times 9\sqrt{2} = 36\sqrt{2}$ mm.

Step6: Find rectangle sides from diagonal

Let short side = $x$, long side = $y$. Diagonal $d=12$ in.
$60^\circ$ angle gives $\sin60^\circ = \frac{y}{d}$, $\cos60^\circ = \frac{x}{d}$.
$y = 12 \times \frac{\sqrt{3}}{2} = 6\sqrt{3}$ in, $x = 12 \times \frac{1}{2}=6$ in.
Perimeter $P=2(x+y)=2(6+6\sqrt{3})=12+12\sqrt{3}$ in.

Answer:

1. $6\sqrt{2}$ in
2. $9$ m
3. $24\sqrt{3}$ ft
4. $\frac{13\sqrt{3}}{2}$ cm
5. $36\sqrt{2}$ mm
6. $12+12\sqrt{3}$ in