The bisector of angle A of rectangle ABCD intersects side BC at point K, BK = 4cm, KC = 8cm. Find the area of the rectangle.

1. Let’s make a drawing.

2. Find the length of the rectangle.

The length of the rectangle BC is equal to the sum of the lengths BK and KC:

BC = BK + KC.

Substitute the known values and find BK:

BC = 4 + 8 = 12 (cm).

3. Determine the width of the rectangle.

The bisector AK bisects the right angle BAD. Respectively,

∠BAK = ∠KAD = 90 °: 2 = 45 °.

Consider a triangle ABK. Him:

∠ABK = 90 °;

∠BAK = 45 °;

∠BKA = 180 ° – (90 ° + 45 °) = 45 °.

Since ∠BAK = ∠BKA, the triangle ABK is isosceles. Means,

AB = BK = 4 cm.

4. Let’s calculate the area of the rectangle.

The area of a rectangle is equal to the product of length and width:

S = BC * AB;

S = 12 * 4;

S = 48 (cm2).