Westonci.ca is the premier destination for reliable answers to your questions, provided by a community of experts. Discover comprehensive solutions to your questions from a wide network of experts on our user-friendly platform. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
Certainly! Let's solve the problem step-by-step. Given the vertices of triangle [tex]\( \triangle ABC \)[/tex]:
- [tex]\( A(2, 8) \)[/tex]
- [tex]\( B(16, 2) \)[/tex]
- [tex]\( C(6, 2) \)[/tex]
### 1. Calculate the lengths of the sides of the triangle
Step 1.1: Calculate the distance [tex]\( AB \)[/tex]
The distance formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
For [tex]\( A(2, 8) \)[/tex] and [tex]\( B(16, 2) \)[/tex]:
[tex]\[ AB = \sqrt{(16 - 2)^2 + (2 - 8)^2} \][/tex]
[tex]\[ AB = \sqrt{14^2 + (-6)^2} \][/tex]
[tex]\[ AB = \sqrt{196 + 36} \][/tex]
[tex]\[ AB = \sqrt{232} \][/tex]
[tex]\[ AB \approx 15.23 \][/tex]
Step 1.2: Calculate the distance [tex]\( BC \)[/tex]
For [tex]\( B(16, 2) \)[/tex] and [tex]\( C(6, 2) \)[/tex]:
[tex]\[ BC = \sqrt{(6 - 16)^2 + (2 - 2)^2} \][/tex]
[tex]\[ BC = \sqrt{(-10)^2 + 0^2} \][/tex]
[tex]\[ BC = \sqrt{100} \][/tex]
[tex]\[ BC = 10.00 \][/tex]
Step 1.3: Calculate the distance [tex]\( AC \)[/tex]
For [tex]\( A(2, 8) \)[/tex] and [tex]\( C(6, 2) \)[/tex]:
[tex]\[ AC = \sqrt{(6 - 2)^2 + (2 - 8)^2} \][/tex]
[tex]\[ AC = \sqrt{4^2 + (-6)^2} \][/tex]
[tex]\[ AC = \sqrt{16 + 36} \][/tex]
[tex]\[ AC = \sqrt{52} \][/tex]
[tex]\[ AC \approx 7.21 \][/tex]
### 2. Calculate the perimeter of the triangle
The perimeter [tex]\( P \)[/tex] of the triangle is the sum of its side lengths:
[tex]\[ P = AB + BC + AC \][/tex]
[tex]\[ P \approx 15.23 + 10.00 + 7.21 \][/tex]
[tex]\[ P \approx 32.44 \][/tex]
### 3. Calculate the area of the triangle
To calculate the area of a triangle given its vertices, use the formula:
[tex]\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \][/tex]
For [tex]\( A(2, 8) \)[/tex], [tex]\( B(16, 2) \)[/tex], and [tex]\( C(6, 2) \)[/tex]:
[tex]\[ \text{Area} = \frac{1}{2} \left| 2(2 - 2) + 16(2 - 8) + 6(8 - 2) \right| \][/tex]
[tex]\[ \text{Area} = \frac{1}{2} \left| 0 + 16(-6) + 6(6) \right| \][/tex]
[tex]\[ \text{Area} = \frac{1}{2} \left| -96 + 36 \right| \][/tex]
[tex]\[ \text{Area} = \frac{1}{2} \left| -60 \right| \][/tex]
[tex]\[ \text{Area} = \frac{1}{2} \times 60 \][/tex]
[tex]\[ \text{Area} = 30.00 \][/tex]
### Final Answer:
Hence, the perimeter of [tex]\( \triangle ABC \)[/tex] is approximately [tex]\( 32.44 \)[/tex] units, and its area is [tex]\( 30.00 \)[/tex] square units.
- [tex]\( A(2, 8) \)[/tex]
- [tex]\( B(16, 2) \)[/tex]
- [tex]\( C(6, 2) \)[/tex]
### 1. Calculate the lengths of the sides of the triangle
Step 1.1: Calculate the distance [tex]\( AB \)[/tex]
The distance formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
For [tex]\( A(2, 8) \)[/tex] and [tex]\( B(16, 2) \)[/tex]:
[tex]\[ AB = \sqrt{(16 - 2)^2 + (2 - 8)^2} \][/tex]
[tex]\[ AB = \sqrt{14^2 + (-6)^2} \][/tex]
[tex]\[ AB = \sqrt{196 + 36} \][/tex]
[tex]\[ AB = \sqrt{232} \][/tex]
[tex]\[ AB \approx 15.23 \][/tex]
Step 1.2: Calculate the distance [tex]\( BC \)[/tex]
For [tex]\( B(16, 2) \)[/tex] and [tex]\( C(6, 2) \)[/tex]:
[tex]\[ BC = \sqrt{(6 - 16)^2 + (2 - 2)^2} \][/tex]
[tex]\[ BC = \sqrt{(-10)^2 + 0^2} \][/tex]
[tex]\[ BC = \sqrt{100} \][/tex]
[tex]\[ BC = 10.00 \][/tex]
Step 1.3: Calculate the distance [tex]\( AC \)[/tex]
For [tex]\( A(2, 8) \)[/tex] and [tex]\( C(6, 2) \)[/tex]:
[tex]\[ AC = \sqrt{(6 - 2)^2 + (2 - 8)^2} \][/tex]
[tex]\[ AC = \sqrt{4^2 + (-6)^2} \][/tex]
[tex]\[ AC = \sqrt{16 + 36} \][/tex]
[tex]\[ AC = \sqrt{52} \][/tex]
[tex]\[ AC \approx 7.21 \][/tex]
### 2. Calculate the perimeter of the triangle
The perimeter [tex]\( P \)[/tex] of the triangle is the sum of its side lengths:
[tex]\[ P = AB + BC + AC \][/tex]
[tex]\[ P \approx 15.23 + 10.00 + 7.21 \][/tex]
[tex]\[ P \approx 32.44 \][/tex]
### 3. Calculate the area of the triangle
To calculate the area of a triangle given its vertices, use the formula:
[tex]\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \][/tex]
For [tex]\( A(2, 8) \)[/tex], [tex]\( B(16, 2) \)[/tex], and [tex]\( C(6, 2) \)[/tex]:
[tex]\[ \text{Area} = \frac{1}{2} \left| 2(2 - 2) + 16(2 - 8) + 6(8 - 2) \right| \][/tex]
[tex]\[ \text{Area} = \frac{1}{2} \left| 0 + 16(-6) + 6(6) \right| \][/tex]
[tex]\[ \text{Area} = \frac{1}{2} \left| -96 + 36 \right| \][/tex]
[tex]\[ \text{Area} = \frac{1}{2} \left| -60 \right| \][/tex]
[tex]\[ \text{Area} = \frac{1}{2} \times 60 \][/tex]
[tex]\[ \text{Area} = 30.00 \][/tex]
### Final Answer:
Hence, the perimeter of [tex]\( \triangle ABC \)[/tex] is approximately [tex]\( 32.44 \)[/tex] units, and its area is [tex]\( 30.00 \)[/tex] square units.
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.
