Question

3.4 cm
3.0 cm
1.6 cm
r
w
48°
x
y
e
w = 1.6
x = 48
y = 1

Explanation:

Step1: Use Pythagorean theorem

In the right - triangle formed by half of the diagonals and the side of the rhombus. For the vertical diagonal part, if the side of the rhombus \(s = 3.4\mathrm{cm}\) and one half - diagonal \(a=1.6\mathrm{cm}\), using the Pythagorean theorem \(c^{2}=a^{2}+b^{2}\), where \(c\) is the side of the rhombus. Let the other half - diagonal (vertical) be \(b\). Then \(b=\sqrt{s^{2}-a^{2}}=\sqrt{3.4^{2}-1.6^{2}}=\sqrt{(3.4 + 1.6)(3.4 - 1.6)}=\sqrt{5\times1.8}=\sqrt{9}=3\mathrm{cm}\). Since the diagonals of a rhombus bisect each other, for \(y\), we know that the diagonals of a rhombus are perpendicular bisectors. The vertical diagonal is composed of two equal parts. Given one part is \(3\mathrm{cm}\), so \(y = 3\mathrm{cm}\)

Step2: Use angle properties

The diagonals of a rhombus bisect the angles. Since the angle adjacent to \(48^{\circ}\) (in the triangle formed by the diagonals and the side of the rhombus) is bisected. The triangles formed by the diagonals of a rhombus are congruent. So \(x = 48^{\circ}\) and \(w=1.6\mathrm{cm}\) (diagonals bisect each other)

Answer:

\(w = 1.6\mathrm{cm}\), \(x = 48^{\circ}\), \(y=3\mathrm{cm}\)