rhombus PQRS such that PQ = 6cm and < SPQ = 75 deg.
Measures the length of PR.
PR = 2 * 6 cm * cos(75º / 2)
because the diagonals are perpendicular and bisect each other.
To find the length of PR in the given rhombus PQRS, we can use the properties of a rhombus.
A rhombus is a quadrilateral in which all four sides are equal. It also has some other special properties:
1. Opposite sides are parallel.
2. Opposite angles are equal.
3. Diagonals bisect each other at right angles.
Now, let's look at the given information. We know that PQ = 6 cm. We need to determine the length of PR.
Since PQRS is a rhombus, all sides are equal. Therefore, SQ = 6 cm as well.
Next, we are given that <SPQ = 75 degrees. As the diagonals bisect each other at right angles in a rhombus, <SPQ is half of a right angle (90 degrees). So, <SPQ = 90/2 = 45 degrees.
Since the opposite angles in a rhombus are equal, <SQR is also 45 degrees.
To find the length of PR, we can use the law of cosines. In triangle PSR, we know two side lengths (PS = 6 cm, SR = PR), and the included angle measure (<PSR = 45 degrees).
The law of cosines states that in a triangle:
c^2 = a^2 + b^2 - 2ab * cos(C)
In our case, a = PS, b = SR = PR, and C = <PSR.
Using the law of cosines, we have:
PR^2 = 6^2 + PR^2 - 2 * 6 * PR * cos(45)
Simplifying the equation:
PR^2 = 36 + PR^2 - 12PR * (1/√2)
PR^2 - PR^2 + 12PR * (1/√2) = 36
12PR * (1/√2) = 36
PR * (√2/2) = 3
PR * √2 = 6
PR = 6 / √2
To rationalize the denominator, we multiply both numerator and denominator by √2:
PR = (6 * √2) / (2)
PR = 3√2 cm
Therefore, the length of PR in rhombus PQRS is 3√2 cm.