Answer :
Certainly! Let’s go through each part of the problem step-by-step.
### Part (a): The angles of a triangle are in the ratio 3:4:3.
#### (i) Angles in degrees:
The sum of the angles in a triangle is 180 degrees.
- The angles are in the ratio 3:4:3.
- The sum of the ratio parts is 3 + 4 + 3 = 10 parts.
Each part is equivalent to:
[tex]\[ \frac{180}{10} = 18 \text{ degrees per part} \][/tex]
Thus, the angles are:
- [tex]\(3 \times 18 = 54 \text{ degrees}\)[/tex]
- [tex]\(4 \times 18 = 72 \text{ degrees}\)[/tex]
- [tex]\(3 \times 18 = 54 \text{ degrees}\)[/tex]
So, the angles in degrees are [tex]\(54\)[/tex] degrees, [tex]\(72\)[/tex] degrees, and [tex]\(54\)[/tex] degrees.
#### (ii) Angles in grades:
To convert degrees to grades:
[tex]\[ 1 \text{ degree} = \frac{200}{180} \text{ grades} \][/tex]
So the angles in grades are:
- [tex]\(54 \times \frac{200}{180} = 60\)[/tex] grades
- [tex]\(72 \times \frac{200}{180} = 80\)[/tex] grades
- [tex]\(54 \times \frac{200}{180} = 60\)[/tex] grades
Thus, the angles in grades are [tex]\(60\)[/tex] grades, [tex]\(80\)[/tex] grades, and [tex]\(60\)[/tex] grades.
#### (iii) Angles in radians:
To convert degrees to radians:
[tex]\[ 1 \text{ degree} = \frac{\pi}{180} \text{ radians} \][/tex]
So the angles in radians are:
- [tex]\(54 \times \frac{\pi}{180} = 0.9424777960769379\)[/tex] radians
- [tex]\(72 \times \frac{\pi}{180} = 1.2566370614359172\)[/tex] radians
- [tex]\(54 \times \frac{\pi}{180} = 0.9424777960769379\)[/tex] radians
Thus, the angles in radians are approximately [tex]\(0.942\)[/tex] radians, [tex]\(1.257\)[/tex] radians, and [tex]\(0.942\)[/tex] radians.
### Part (b): The angles of a quadrilateral are in the ratio 2:3:4:1.
The sum of the angles in a quadrilateral is 360 degrees.
#### (i) Angles in degrees:
- The angles are in the ratio 2:3:4:1.
- The sum of the ratio parts is 2 + 3 + 4 + 1 = 10 parts.
Each part is equivalent to:
[tex]\[ \frac{360}{10} = 36 \text{ degrees per part} \][/tex]
Thus, the angles are:
- [tex]\(2 \times 36 = 72 \text{ degrees}\)[/tex]
- [tex]\(3 \times 36 = 108 \text{ degrees}\)[/tex]
- [tex]\(4 \times 36 = 144 \text{ degrees}\)[/tex]
- [tex]\(1 \times 36 = 36 \text{ degrees}\)[/tex]
So, the angles in degrees are [tex]\(72\)[/tex] degrees, [tex]\(108\)[/tex] degrees, [tex]\(144\)[/tex] degrees, and [tex]\(36\)[/tex] degrees.
#### (ii) Angles in grades:
To convert degrees to grades:
[tex]\[ 1 \text{ degree} = \frac{200}{180} \text{ grades} \][/tex]
So the angles in grades are:
- [tex]\(72 \times \frac{200}{180} = 80\)[/tex] grades
- [tex]\(108 \times \frac{200}{180} = 120\)[/tex] grades
- [tex]\(144 \times \frac{200}{180} = 160\)[/tex] grades
- [tex]\(36 \times \frac{200}{180} = 40\)[/tex] grades
Thus, the angles in grades are [tex]\(80\)[/tex] grades, [tex]\(120\)[/tex] grades, [tex]\(160\)[/tex] grades, and [tex]\(40\)[/tex] grades.
#### (iii) Angles in radians:
To convert degrees to radians:
[tex]\[ 1 \text{ degree} = \frac{\pi}{180} \text{ radians} \][/tex]
So the angles in radians are:
- [tex]\(72 \times \frac{\pi}{180} = 1.2566370614359172\)[/tex] radians
- [tex]\(108 \times \frac{\pi}{180} = 1.8849555921538759\)[/tex] radians
- [tex]\(144 \times \frac{\pi}{180} = 2.5132741228718345\)[/tex] radians
- [tex]\(36 \times \frac{\pi}{180} = 0.6283185307179586\)[/tex] radians
Thus, the angles in radians are approximately [tex]\(1.257\)[/tex] radians, [tex]\(1.885\)[/tex] radians, [tex]\(2.513\)[/tex] radians, and [tex]\(0.628\)[/tex] radians.
### Part (c): The two angles of a triangle are in the ratio 4:5 and the third angle is 90 degrees.
#### Angles in grades:
The sum of the angles in a triangle is 180 degrees.
- Given the third angle is 90 degrees, the remaining two angles sum up to [tex]\(180 - 90 = 90\)[/tex] degrees.
- The two angles are in the ratio 4:5.
- The sum of the ratio parts is 4 + 5 = 9 parts.
Each part is equivalent to:
[tex]\[ \frac{90}{9} = 10 \text{ degrees per part} \][/tex]
Thus, the angles are:
- [tex]\(4 \times 10 = 40 \text{ degrees}\)[/tex]
- [tex]\(5 \times 10 = 50 \text{ degrees}\)[/tex]
- The third angle is [tex]\(90 \text{ degrees}\)[/tex]
To convert degrees to grades:
[tex]\[ 1 \text{ degree} = \frac{200}{180} \text{ grades} \][/tex]
So the angles in grades are:
- [tex]\(40 \times \frac{200}{180} = 44.44444444444444\)[/tex] grades
- [tex]\(50 \times \frac{200}{180} = 55.55555555555556\)[/tex] grades
- [tex]\(90 \times \frac{200}{180} = 100\)[/tex] grades
Thus, the angles in grades are [tex]\(44.44\)[/tex] grades, [tex]\(55.56\)[/tex] grades, and [tex]\(100\)[/tex] grades.
This completes our detailed step-by-step solution for the given problem.
### Part (a): The angles of a triangle are in the ratio 3:4:3.
#### (i) Angles in degrees:
The sum of the angles in a triangle is 180 degrees.
- The angles are in the ratio 3:4:3.
- The sum of the ratio parts is 3 + 4 + 3 = 10 parts.
Each part is equivalent to:
[tex]\[ \frac{180}{10} = 18 \text{ degrees per part} \][/tex]
Thus, the angles are:
- [tex]\(3 \times 18 = 54 \text{ degrees}\)[/tex]
- [tex]\(4 \times 18 = 72 \text{ degrees}\)[/tex]
- [tex]\(3 \times 18 = 54 \text{ degrees}\)[/tex]
So, the angles in degrees are [tex]\(54\)[/tex] degrees, [tex]\(72\)[/tex] degrees, and [tex]\(54\)[/tex] degrees.
#### (ii) Angles in grades:
To convert degrees to grades:
[tex]\[ 1 \text{ degree} = \frac{200}{180} \text{ grades} \][/tex]
So the angles in grades are:
- [tex]\(54 \times \frac{200}{180} = 60\)[/tex] grades
- [tex]\(72 \times \frac{200}{180} = 80\)[/tex] grades
- [tex]\(54 \times \frac{200}{180} = 60\)[/tex] grades
Thus, the angles in grades are [tex]\(60\)[/tex] grades, [tex]\(80\)[/tex] grades, and [tex]\(60\)[/tex] grades.
#### (iii) Angles in radians:
To convert degrees to radians:
[tex]\[ 1 \text{ degree} = \frac{\pi}{180} \text{ radians} \][/tex]
So the angles in radians are:
- [tex]\(54 \times \frac{\pi}{180} = 0.9424777960769379\)[/tex] radians
- [tex]\(72 \times \frac{\pi}{180} = 1.2566370614359172\)[/tex] radians
- [tex]\(54 \times \frac{\pi}{180} = 0.9424777960769379\)[/tex] radians
Thus, the angles in radians are approximately [tex]\(0.942\)[/tex] radians, [tex]\(1.257\)[/tex] radians, and [tex]\(0.942\)[/tex] radians.
### Part (b): The angles of a quadrilateral are in the ratio 2:3:4:1.
The sum of the angles in a quadrilateral is 360 degrees.
#### (i) Angles in degrees:
- The angles are in the ratio 2:3:4:1.
- The sum of the ratio parts is 2 + 3 + 4 + 1 = 10 parts.
Each part is equivalent to:
[tex]\[ \frac{360}{10} = 36 \text{ degrees per part} \][/tex]
Thus, the angles are:
- [tex]\(2 \times 36 = 72 \text{ degrees}\)[/tex]
- [tex]\(3 \times 36 = 108 \text{ degrees}\)[/tex]
- [tex]\(4 \times 36 = 144 \text{ degrees}\)[/tex]
- [tex]\(1 \times 36 = 36 \text{ degrees}\)[/tex]
So, the angles in degrees are [tex]\(72\)[/tex] degrees, [tex]\(108\)[/tex] degrees, [tex]\(144\)[/tex] degrees, and [tex]\(36\)[/tex] degrees.
#### (ii) Angles in grades:
To convert degrees to grades:
[tex]\[ 1 \text{ degree} = \frac{200}{180} \text{ grades} \][/tex]
So the angles in grades are:
- [tex]\(72 \times \frac{200}{180} = 80\)[/tex] grades
- [tex]\(108 \times \frac{200}{180} = 120\)[/tex] grades
- [tex]\(144 \times \frac{200}{180} = 160\)[/tex] grades
- [tex]\(36 \times \frac{200}{180} = 40\)[/tex] grades
Thus, the angles in grades are [tex]\(80\)[/tex] grades, [tex]\(120\)[/tex] grades, [tex]\(160\)[/tex] grades, and [tex]\(40\)[/tex] grades.
#### (iii) Angles in radians:
To convert degrees to radians:
[tex]\[ 1 \text{ degree} = \frac{\pi}{180} \text{ radians} \][/tex]
So the angles in radians are:
- [tex]\(72 \times \frac{\pi}{180} = 1.2566370614359172\)[/tex] radians
- [tex]\(108 \times \frac{\pi}{180} = 1.8849555921538759\)[/tex] radians
- [tex]\(144 \times \frac{\pi}{180} = 2.5132741228718345\)[/tex] radians
- [tex]\(36 \times \frac{\pi}{180} = 0.6283185307179586\)[/tex] radians
Thus, the angles in radians are approximately [tex]\(1.257\)[/tex] radians, [tex]\(1.885\)[/tex] radians, [tex]\(2.513\)[/tex] radians, and [tex]\(0.628\)[/tex] radians.
### Part (c): The two angles of a triangle are in the ratio 4:5 and the third angle is 90 degrees.
#### Angles in grades:
The sum of the angles in a triangle is 180 degrees.
- Given the third angle is 90 degrees, the remaining two angles sum up to [tex]\(180 - 90 = 90\)[/tex] degrees.
- The two angles are in the ratio 4:5.
- The sum of the ratio parts is 4 + 5 = 9 parts.
Each part is equivalent to:
[tex]\[ \frac{90}{9} = 10 \text{ degrees per part} \][/tex]
Thus, the angles are:
- [tex]\(4 \times 10 = 40 \text{ degrees}\)[/tex]
- [tex]\(5 \times 10 = 50 \text{ degrees}\)[/tex]
- The third angle is [tex]\(90 \text{ degrees}\)[/tex]
To convert degrees to grades:
[tex]\[ 1 \text{ degree} = \frac{200}{180} \text{ grades} \][/tex]
So the angles in grades are:
- [tex]\(40 \times \frac{200}{180} = 44.44444444444444\)[/tex] grades
- [tex]\(50 \times \frac{200}{180} = 55.55555555555556\)[/tex] grades
- [tex]\(90 \times \frac{200}{180} = 100\)[/tex] grades
Thus, the angles in grades are [tex]\(44.44\)[/tex] grades, [tex]\(55.56\)[/tex] grades, and [tex]\(100\)[/tex] grades.
This completes our detailed step-by-step solution for the given problem.
