Join the IDNLearn.com community and get your questions answered by experts. Our experts provide accurate and detailed responses to help you navigate any topic or issue with confidence.
Sagot :
Certainly! Let's break down each part of the question:
### Part 1: Find all angles of the first triangle
We are given that two angles of a triangle are in the ratio [tex]\(5:4\)[/tex] and the third angle is [tex]\(90^\circ\)[/tex].
1. Recall that the sum of all angles in any triangle is [tex]\(180^\circ\)[/tex].
2. Let the two unknown angles be represented as [tex]\(5x\)[/tex] and [tex]\(4x\)[/tex]. The third angle is [tex]\(90^\circ\)[/tex].
3. The sum of all angles in the triangle:
[tex]\[ 5x + 4x + 90^\circ = 180^\circ \][/tex]
4. Combine like terms:
[tex]\[ 9x + 90^\circ = 180^\circ \][/tex]
5. Subtract [tex]\(90^\circ\)[/tex] from both sides:
[tex]\[ 9x = 90^\circ \][/tex]
6. Divide by [tex]\(9\)[/tex]:
[tex]\[ x = 10^\circ \][/tex]
7. Now, calculate the actual angles:
[tex]\[ 5x = 5 \times 10^\circ = 50^\circ \][/tex]
[tex]\[ 4x = 4 \times 10^\circ = 40^\circ \][/tex]
So, the angles of the first triangle are:
[tex]\[ 50^\circ, 40^\circ, 90^\circ \][/tex]
### Part 2: Find all angles of the second triangle
We are given that one angle of a triangle is [tex]\(50^\circ\)[/tex] and the ratio of the remaining two angles is [tex]\(7:8\)[/tex].
1. Recall again that the sum of all angles in a triangle is [tex]\(180^\circ\)[/tex].
2. Let the remaining two angles be represented as [tex]\(7y\)[/tex] and [tex]\(8y\)[/tex]. The given angle is [tex]\(50^\circ\)[/tex].
3. The sum of all angles in the triangle:
[tex]\[ 50^\circ + 7y + 8y = 180^\circ \][/tex]
4. Combine like terms:
[tex]\[ 50^\circ + 15y = 180^\circ \][/tex]
5. Subtract [tex]\(50^\circ\)[/tex] from both sides:
[tex]\[ 15y = 130^\circ \][/tex]
6. Divide by [tex]\(15\)[/tex]:
[tex]\[ y \approx 8.6667^\circ \][/tex]
7. Now, calculate the actual angles:
[tex]\[ 7y = 7 \times 8.6667^\circ \approx 60.6667^\circ \][/tex]
[tex]\[ 8y = 8 \times 8.6667^\circ \approx 69.3333^\circ \][/tex]
So, the angles of the second triangle are:
[tex]\[ 50^\circ, 60.6667^\circ, 69.3333^\circ \][/tex]
In summary, the angles are:
1. For the first triangle: [tex]\(50^\circ, 40^\circ, 90^\circ\)[/tex]
2. For the second triangle: [tex]\(50^\circ, 60.6667^\circ, 69.3333^\circ\)[/tex]
### Part 1: Find all angles of the first triangle
We are given that two angles of a triangle are in the ratio [tex]\(5:4\)[/tex] and the third angle is [tex]\(90^\circ\)[/tex].
1. Recall that the sum of all angles in any triangle is [tex]\(180^\circ\)[/tex].
2. Let the two unknown angles be represented as [tex]\(5x\)[/tex] and [tex]\(4x\)[/tex]. The third angle is [tex]\(90^\circ\)[/tex].
3. The sum of all angles in the triangle:
[tex]\[ 5x + 4x + 90^\circ = 180^\circ \][/tex]
4. Combine like terms:
[tex]\[ 9x + 90^\circ = 180^\circ \][/tex]
5. Subtract [tex]\(90^\circ\)[/tex] from both sides:
[tex]\[ 9x = 90^\circ \][/tex]
6. Divide by [tex]\(9\)[/tex]:
[tex]\[ x = 10^\circ \][/tex]
7. Now, calculate the actual angles:
[tex]\[ 5x = 5 \times 10^\circ = 50^\circ \][/tex]
[tex]\[ 4x = 4 \times 10^\circ = 40^\circ \][/tex]
So, the angles of the first triangle are:
[tex]\[ 50^\circ, 40^\circ, 90^\circ \][/tex]
### Part 2: Find all angles of the second triangle
We are given that one angle of a triangle is [tex]\(50^\circ\)[/tex] and the ratio of the remaining two angles is [tex]\(7:8\)[/tex].
1. Recall again that the sum of all angles in a triangle is [tex]\(180^\circ\)[/tex].
2. Let the remaining two angles be represented as [tex]\(7y\)[/tex] and [tex]\(8y\)[/tex]. The given angle is [tex]\(50^\circ\)[/tex].
3. The sum of all angles in the triangle:
[tex]\[ 50^\circ + 7y + 8y = 180^\circ \][/tex]
4. Combine like terms:
[tex]\[ 50^\circ + 15y = 180^\circ \][/tex]
5. Subtract [tex]\(50^\circ\)[/tex] from both sides:
[tex]\[ 15y = 130^\circ \][/tex]
6. Divide by [tex]\(15\)[/tex]:
[tex]\[ y \approx 8.6667^\circ \][/tex]
7. Now, calculate the actual angles:
[tex]\[ 7y = 7 \times 8.6667^\circ \approx 60.6667^\circ \][/tex]
[tex]\[ 8y = 8 \times 8.6667^\circ \approx 69.3333^\circ \][/tex]
So, the angles of the second triangle are:
[tex]\[ 50^\circ, 60.6667^\circ, 69.3333^\circ \][/tex]
In summary, the angles are:
1. For the first triangle: [tex]\(50^\circ, 40^\circ, 90^\circ\)[/tex]
2. For the second triangle: [tex]\(50^\circ, 60.6667^\circ, 69.3333^\circ\)[/tex]
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! For clear and precise answers, choose IDNLearn.com. Thanks for stopping by, and come back soon for more valuable insights.
