Triangles BAD and BDC are right triangles with AB = 12 units, BD = 15 units, and $BC = 17 units. What is the area, in square units, of quadrilateral ABCD?

The area of the quadrilateral ABCD is 114 square units

Step-by-step explanation:

We must calculate the area of each triangle and then add these areas so we calculate the area of the quadrilateral ABCD

First for the BAD right triangle:

AD = sqrt [BD ^ 2 - AB ^ 2]

AD = sqrt [15 ^ 2 - 12 ^ 2]

AD = sqrt [225-144]

AD = sqrt [81]

AD = 9

The area of a triangle is half the product of the base times the height, that is:

A1 = AB * AD / 2 = 12 * 9/2 = 54

Then for the second triangle in the right triangle BDC:

DC = sqrt [BC ^ 2 - BD ^ 2]

DC = sqrt [17 ^ 2 - 15 ^ 2]

DC = sqrt [289 - 225]

DC = sqrt [64] = 8

We calculate the area

A2 = DC * BD / 2 = 8 * 15/2 = 60

The total area then is:

AT = A1 + A2

AT = 54 + 60 = 114

Which means that the area of the quadrilateral ABCD is 114 square units