In the figure AB = 11cm, BC = 8cm, AD = 3cm, AC = 5cm and DAC is a right angle.
The size of ADC;
The size of ACB;
The area of the quadrilateral ABCD.
tan ADC = 5/3
11^2 = 5^2+8^2 - 2*5*8 cos ACB
Now you can figure the areas of the triangles ACD and ACB, and just add them up for the area of ABCD
So aera =1/2(AB)(BC)sin(ACB)
To find the angles and area of the given shape, we can use basic trigonometric principles and the formula for the area of a quadrilateral.
1. Angle ADC:
In triangle ADC, we have a right angle at D. We can use the Pythagorean theorem to find the length of CD:
CD^2 = AC^2 - AD^2
CD^2 = 5^2 - 3^2
CD^2 = 25 - 9
CD^2 = 16
CD = √16
CD = 4
Now, we can use the cosine rule to find the size of angle ADC:
cos(ADC) = (AC^2 + CD^2 - AD^2) / (2 * AC * CD)
cos(ADC) = (5^2 + 4^2 - 3^2) / (2 * 5 * 4)
cos(ADC) = (25 + 16 - 9) / 40
cos(ADC) = 32 / 40
cos(ADC) = 0.8
To find the size of angle ADC, we can take the arccosine of 0.8:
ADC = arccos(0.8)
ADC ≈ 38.69 degrees
2. Angle ACB:
To find angle ACB, we can use the sine rule, as we have two known sides and the included angle:
sin(ACB) / BC = sin(ADC) / AC
sin(ACB) / 8 = sin(38.69) / 5
Now, we can solve for sin(ACB):
sin(ACB) = (8 * sin(38.69)) / 5
sin(ACB) ≈ 0.783
To find the size of angle ACB, we can take the arcsine of 0.783:
ACB = arcsin(0.783)
ACB ≈ 51.31 degrees
3. Area of ABCD:
To find the area of the quadrilateral ABCD, we can split it into two triangles: triangle ABC and triangle ADC. The total area will be the sum of the areas of these two triangles.
Area of triangle ABC:
We can use Heron's formula to find the area of ABC, as we know all three sides:
s = (AB + BC + AC) / 2
s = (11 + 8 + 5) / 2
s = 24 / 2
s = 12
Area_ABC = √(s * (s - AB) * (s - BC) * (s - AC))
Area_ABC = √(12 * (12 - 11) * (12 - 8) * (12 - 5))
Area_ABC = √(12 * 1 * 4 * 7)
Area_ABC = √(336)
Area_ABC ≈ 18.33 square cm
Area of triangle ADC:
We can find the area of ADC using the formula for the area of a right-angled triangle:
Area_ADC = (AC * AD) / 2
Area_ADC = (5 * 3) / 2
Area_ADC = 15 / 2
Area_ADC = 7.5 square cm
Total area of quadrilateral ABCD = Area_ABC + Area_ADC
Total area of quadrilateral ABCD ≈ 18.33 + 7.5
Total area of quadrilateral ABCD ≈ 25.83 square cm