In the figure AB = 11cm, BC = 8cm, AD = 3cm, AC = 5cm and DAC is a right angle.

(a) length DC;

(b) the size of ADC;

(c) the size of ACB;

(d) the area of the quadrilateral ABCD.

DC is the hypotenuse of a right triangle, so that's easy.

tan ∠ADC = 5/3
use the law of cosines to find ∠ACB

If you know two sides A and B of a triangle, and the angle θ between them, then the area of the triangle is 1/2 AB sinθ

The area of ABCD is the sum of triangles ABD and ADC

To find the answers to these questions, we need to apply some concepts from geometry and trigonometry. Let's break down each question and explain how to solve them.

(a) Length DC:
To find the length of DC, we can use the Pythagorean theorem, as DAC is a right angle. According to the Pythagorean theorem, in a right-angled triangle, the square of the hypotenuse (AC) is equal to the sum of the squares of the two other sides (AD and DC):
AC^2 = AD^2 + DC^2

In this case, AC = 5cm and AD = 3cm. Plugging these values into the equation, we can solve for DC:

5^2 = 3^2 + DC^2
25 = 9 + DC^2
DC^2 = 25 - 9
DC^2 = 16
DC = sqrt(16)
DC = 4cm

Therefore, the length of DC is 4cm.

(b) Size of ADC:
To find the size of ADC, we can use the trigonometric function tangent (or inverse tangent). The tangent of an angle in a right-angled triangle is defined as the ratio of the length of the side opposite the angle (AD) to the length of the side adjacent to the angle (DC):
tan(ADC) = AD / DC

In this case, AD = 3cm and DC = 4cm. Plugging these values into the equation, we can solve for ADC:

tan(ADC) = 3 / 4
ADC = atan(3 / 4)

Using a calculator or trigonometric tables, we find that atan(3 / 4) is approximately 36.87 degrees.

Therefore, the size of ADC is approximately 36.87 degrees.

(c) Size of ACB:
To find the size of ACB, we can use the trigonometric function sine (or cosine, since they are complementary angles) in triangle ABC:
sin(ACB) = BC / AC

In this case, BC = 8cm and AC = 5cm. Plugging these values into the equation, we can solve for ACB:

sin(ACB) = 8 / 5
ACB = asin(8 / 5)

Using a calculator or trigonometric tables, we find that asin(8 / 5) is approximately 53.13 degrees.

Therefore, the size of ACB is approximately 53.13 degrees.

(d) Area of the quadrilateral ABCD:
To find the area of the quadrilateral ABCD, we need to divide it into two triangles (ADC and ACB) and then find the areas of each triangle separately. Then, we can sum up these two areas.

Area(ADCB) = Area(ADC) + Area(ACB)

We have already determined the sizes of ADC and ACB, so we can now calculate the areas of each triangle using the formula:

Area = (1/2) * base * height

For triangle ADC:
Area(ADC) = (1/2) * DC * AD

And for triangle ACB:
Area(ACB) = (1/2) * AC * BC

Plugging in the known values:
Area(ADC) = (1/2) * 4 * 3 = 6 square cm
Area(ACB) = (1/2) * 5 * 8 = 20 square cm

Finally, we sum up the areas of both triangles:
Area(ADCB) = 6 + 20 = 26 square cm

Therefore, the area of the quadrilateral ABCD is 26 square cm.