Find the perimeter and area of the rhombus if the diagonals are 8 and 9 cm.

Given:

a = 8 cm;

b = 9 cm;

P =? cm.

S =? cm².

The rhombus is divided by diagonals into 4 equal right-angled triangles. Let’s calculate their legs:

1) 8: 2 = 4 (cm)

2) 9: 2 = 4.5 (cm).

We got two such triangles.

a1 = 8 cm; a2 = 8 cm;
b1 = 9 cm; b2 = 9 cm;

with 1 = 4 cm; with 2 = 45 cm;
Let’s calculate their area, apply Heron’s formula S = √p (p – a) * (p – b) * (p – c).

Find the semiperimeter of the first triangle by the formula p = (a + b + c): 2.

1) p1 = (8 + 9 + 4): 2 = 21: 2 = 10.5 (cm).

Find the semi-perimeter of the first triangle:

2) p2 = (8 + 9 + 4.5): 2 = 21.5: 2 = 10.75 (cm).

3) p – a = 10.5 – 8 = 2.5;

p – b = 10.5 – 9 = 1.5;

p – c = 10.5 – 4 = 6.5;

3) p – a = 10.75 – 8 = 2.75;

p – b = 10.75 – 9 = 1.75;

p – c = 10.75 – 4 = 6.75;

5) S1 = √10.5 * 1.5 * 6.5 * 2.5 = √255 = 16;

S2 = √10.5 * 2.75 * 1.75 * 6.75 = √341 = 18.5;

4) S = 2 * S1 + 2 * S2 = 2 * 16 + 2 * 18.5 = 32 + 37 = 69 (cm²).
Answer: 69 cm² area of ​​a rhombus.