Question
athan draws a triangle on the coordinate grid. He marks his points at (1,5)
(
1
,
5
)
, (4,1)
(
4
,
1
)
, and (−4,−5)
(
−
4
,
−
5
)
. What is the perimeter of the triangle, rounded to the nearest tenth of a unit? Explain the steps that Nathan must take to find the solution.
this is the question
To find the perimeter of the triangle at (1,5) , (4,1) , and (-4,-5) , calculate the lengths of each side using the distance formula. The side lengths are approximately 5, 10, and 11.2 units. Adding these gives a perimeter of 26.2 units, rounded to the nearest tenth.
this is the answer
but they want me to ad to my answer this is the question
I need to see setups and steps as to how you found the lengths of each side
(
1
,
5
)
, (4,1)
(
4
,
1
)
, and (−4,−5)
(
−
4
,
−
5
)
. What is the perimeter of the triangle, rounded to the nearest tenth of a unit? Explain the steps that Nathan must take to find the solution.
this is the question
To find the perimeter of the triangle at (1,5) , (4,1) , and (-4,-5) , calculate the lengths of each side using the distance formula. The side lengths are approximately 5, 10, and 11.2 units. Adding these gives a perimeter of 26.2 units, rounded to the nearest tenth.
this is the answer
but they want me to ad to my answer this is the question
I need to see setups and steps as to how you found the lengths of each side
Answers
Answered by
GPT-4o mini
To find the perimeter of the triangle formed by the points \( A(1, 5) \), \( B(4, 1) \), and \( C(-4, -5) \), Nathan can follow these steps:
### Step 1: Use the Distance Formula
The distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) on a coordinate grid is given by the formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
### Step 2: Calculate Lengths of Each Side
#### Side AB
- Points: \( A(1, 5) \) and \( B(4, 1) \)
Plugging into the distance formula:
\[
AB = \sqrt{(4 - 1)^2 + (1 - 5)^2}
\]
\[
= \sqrt{(3)^2 + (-4)^2}
\]
\[
= \sqrt{9 + 16}
\]
\[
= \sqrt{25}
\]
\[
= 5
\]
#### Side BC
- Points: \( B(4, 1) \) and \( C(-4, -5) \)
Using the distance formula:
\[
BC = \sqrt{(-4 - 4)^2 + (-5 - 1)^2}
\]
\[
= \sqrt{(-8)^2 + (-6)^2}
\]
\[
= \sqrt{64 + 36}
\]
\[
= \sqrt{100}
\]
\[
= 10
\]
#### Side CA
- Points: \( C(-4, -5) \) and \( A(1, 5) \)
Using the distance formula:
\[
CA = \sqrt{(1 - (-4))^2 + (5 - (-5))^2}
\]
\[
= \sqrt{(1 + 4)^2 + (5 + 5)^2}
\]
\[
= \sqrt{(5)^2 + (10)^2}
\]
\[
= \sqrt{25 + 100}
\]
\[
= \sqrt{125}
\]
\[
= 5\sqrt{5} \approx 11.2 \quad (\text{since } \sqrt{5} \approx 2.236)
\]
### Step 3: Sum the Lengths to Find the Perimeter
Now that we have the lengths of each side:
- \( AB = 5 \)
- \( BC = 10 \)
- \( CA \approx 11.2 \)
Adding these lengths together:
\[
\text{Perimeter} = AB + BC + CA
\]
\[
= 5 + 10 + 11.2
\]
\[
= 26.2
\]
### Final Answer
The perimeter of the triangle, rounded to the nearest tenth of a unit, is approximately **26.2** units.
By following these detailed steps, Nathan can confidently compute the perimeter of the triangle formed by the three given points on the coordinate grid.
### Step 1: Use the Distance Formula
The distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) on a coordinate grid is given by the formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
### Step 2: Calculate Lengths of Each Side
#### Side AB
- Points: \( A(1, 5) \) and \( B(4, 1) \)
Plugging into the distance formula:
\[
AB = \sqrt{(4 - 1)^2 + (1 - 5)^2}
\]
\[
= \sqrt{(3)^2 + (-4)^2}
\]
\[
= \sqrt{9 + 16}
\]
\[
= \sqrt{25}
\]
\[
= 5
\]
#### Side BC
- Points: \( B(4, 1) \) and \( C(-4, -5) \)
Using the distance formula:
\[
BC = \sqrt{(-4 - 4)^2 + (-5 - 1)^2}
\]
\[
= \sqrt{(-8)^2 + (-6)^2}
\]
\[
= \sqrt{64 + 36}
\]
\[
= \sqrt{100}
\]
\[
= 10
\]
#### Side CA
- Points: \( C(-4, -5) \) and \( A(1, 5) \)
Using the distance formula:
\[
CA = \sqrt{(1 - (-4))^2 + (5 - (-5))^2}
\]
\[
= \sqrt{(1 + 4)^2 + (5 + 5)^2}
\]
\[
= \sqrt{(5)^2 + (10)^2}
\]
\[
= \sqrt{25 + 100}
\]
\[
= \sqrt{125}
\]
\[
= 5\sqrt{5} \approx 11.2 \quad (\text{since } \sqrt{5} \approx 2.236)
\]
### Step 3: Sum the Lengths to Find the Perimeter
Now that we have the lengths of each side:
- \( AB = 5 \)
- \( BC = 10 \)
- \( CA \approx 11.2 \)
Adding these lengths together:
\[
\text{Perimeter} = AB + BC + CA
\]
\[
= 5 + 10 + 11.2
\]
\[
= 26.2
\]
### Final Answer
The perimeter of the triangle, rounded to the nearest tenth of a unit, is approximately **26.2** units.
By following these detailed steps, Nathan can confidently compute the perimeter of the triangle formed by the three given points on the coordinate grid.