Perpendicular drawn from the vertex of a right triangle to the diagonal. divides it into

Perpendicular drawn from the vertex of a right triangle to the diagonal. divides it into segments equal to 2 cm and 8 cm. Find the area of the rectangle.

Since the perpendicular divides the diagonal into segments of 2 and 8 cm, the length of the diagonal is:

d = 2 + 8 = 10 cm.

The segments into which the perpendicular h drawn from the vertex divides the diagonal are the projections of the sides a and b of the rectangle onto this diagonal. For two right-angled triangles formed by sides a and b, their projections and perpendicular h, we can write:

h ^ 2 = a ^ 2 – 2 ^ 2 and h ^ 2 = b ^ 2 – 8 ^ 2.

Hence:

a ^ 2 – 2 ^ 2 = b ^ 2 – 8 ^ 2;

a ^ 2 – 4 = b ^ 2 – 64.

For a right-angled triangle formed by adjacent sides a and b and diagonal d, also by the Pythagorean theorem:

a ^ 2 + b ^ 2 = 10 ^ 2;

a ^ 2 = 100 – b ^ 2;

Substitute the resulting expression for a ^ 2 into the equation a ^ 2 – 4 = b ^ 2 – 64.

100 – b ^ 2 – 4 = b ^ 2 – 64;

2 * b ^ 2 = 160;

b ^ 2 = 80;

b = √80 = 4√5 cm.

a ^ 2 = 100 – b ^ 2 = 100 – 80 = 20;

a = √20 = 2√5 cm.

The area of ​​a rectangle is equal to the product of its sides a and b:

S = a * b = 2√5 * 4√5 = 40 cm2.