Question

finding the perimeter of an isosceles triangle of known area
the flag is an isosceles triangle.
the bottom side of the flag measures 5 feet because
the triangle is ⋄. because
⋄, the
trigonometric triangle formula is used to calculate that
the area is $\frac{1}{2}(5)(5)\sin(36^{\circ}) \approx 7.35$ square feet.
recalling that this triangle is isosceles, we can write
the area equation to solve for $x$.
⋄ = $\frac{1}{2}(x)(5)\sin(\diamond)$.
solving for $x$ and adding this value to the length of the
other sides, the perimeter ≈ ⋄ (rounded to the nearest tenth).

Explanation:

Step1: Identify triangle property

The flag is an isosceles triangle, so the bottom side = 5 ft (equal to the given side). The angle between the two equal sides is not 36°; the 36° is the vertex angle at the tip, so the angle between side $x$ and the 5 ft side is $180^\circ - 36^\circ = 144^\circ$.

Step2: Set up area equation

We know the area is $\approx7.35$ sq ft. Use the trigonometric area formula:
$$7.35 = \frac{1}{2}(x)(5)\sin(144^\circ)$$

Step3: Solve for $x$

First, calculate $\sin(144^\circ) \approx 0.5878$. Rearrange the equation:
$$x = \frac{2 \times 7.35}{5 \times 0.5878}$$
$$x = \frac{14.7}{2.939} \approx 5.0$$

Step4: Calculate perimeter

Add all sides: $P = x + 5 + 5$
$$P \approx 5.0 + 5 + 5 = 15.0$$

Answer:

The filled blanks are:

1. isosceles
2. we know two sides and included angle
3. $7.35$
4. $144^\circ$
5. $15.0$