A rhombus with one diagonal equal to the side is equal to an isosceles

A rhombus with one diagonal equal to the side is equal to an isosceles right-angled triangle with hypotenuse 8. Find the square of the side of this rhombus.

1. Vertices of the triangle M, K, T. ∠A = 90 °. MK = MT. KT = 8 units of measurement. The vertices of the rhombus A, B, C, D. S – area.

2. AT² + MK² = 8² = 64.

2AT² = 64.

AT = √32 = 4√2 units of measurement.

3. S ∆ MKT = AT x MK / 2 = 4√2 x 4√2 / 2 = 16 units of measurement².

4. S ∆ MKT is equal to S rhombus = 16 units of measurement².

5. The diagonal ВD of the rhombus divides it into equal ∆АВD and ∆BCD.

6. Triangles are equilateral, since the length of the diagonal is equal to the side of the rhombus.

7. S ∆ ABD = AD²√3 / 4 = 16/2.

AD² = 32√3 / 3 units.