Question

using algebra find the value of each variable. write answers in simplest radical form. 20. diagram of a rhombus with a diagonal of (6sqrt{2}), angles of (45^circ), and sides labeled (n) with right angles at some vertices

Explanation:

Step1: Identify triangle type

The figure is a rhombus (or a kite with two 45° angles, forming a right isosceles triangle when split by the diagonal). The diagonal $6\sqrt{2}$ is the hypotenuse of a right isosceles triangle with legs $n$.

Step2: Apply Pythagorean theorem

For a right isosceles triangle, $n^2 + n^2 = (6\sqrt{2})^2$.
Simplify the equation:
$$2n^2 = 36 \times 2$$
$$2n^2 = 72$$

Step3: Solve for $n$

Divide both sides by 2:
$$n^2 = 36$$
Take the square root:
$$n = \sqrt{36} = 6$$

Answer:

$n = 6$