Question

Nathan draws a triangle on the coordinate grid. He marks his points at (1,5) , (4,1) , and (−4,−5) . What is the perimeter of the triangle, rounded to the nearest tenth of a unit? Show all steps that Nathan must take to find the solution.(4 points)

Answer 1

To find the perimeter of the triangle formed by the points \((1,5)\), \((4,1)\), and \((-4,-5)\), we first need to calculate the lengths of each side of the triangle using the distance formula, which is given by:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

### Step 1: Calculate the distance between points \((1,5)\) and \((4,1)\)

Let \((x_1, y_1) = (1, 5)\) and \((x_2, y_2) = (4, 1)\).

\[
d_{12} = \sqrt{(4 - 1)^2 + (1 - 5)^2}
\]
\[
= \sqrt{(3)^2 + (-4)^2}
\]
\[
= \sqrt{9 + 16}
\]
\[
= \sqrt{25}
\]
\[
= 5
\]

### Step 2: Calculate the distance between points \((4,1)\) and \((-4,-5)\)

Let \((x_1, y_1) = (4, 1)\) and \((x_2, y_2) = (-4, -5)\).

\[
d_{23} = \sqrt{(-4 - 4)^2 + (-5 - 1)^2}
\]
\[
= \sqrt{(-8)^2 + (-6)^2}
\]
\[
= \sqrt{64 + 36}
\]
\[
= \sqrt{100}
\]
\[
= 10
\]

### Step 3: Calculate the distance between points \((-4,-5)\) and \((1,5)\)

Let \((x_1, y_1) = (-4, -5)\) and \((x_2, y_2) = (1, 5)\).

\[
d_{31} = \sqrt{(1 - (-4))^2 + (5 - (-5))^2}
\]
\[
= \sqrt{(1 + 4)^2 + (5 + 5)^2}
\]
\[
= \sqrt{(5)^2 + (10)^2}
\]
\[
= \sqrt{25 + 100}
\]
\[
= \sqrt{125}
\]
\[
= 5\sqrt{5}
\]

To get an approximate value for \(5\sqrt{5}\):

\[
\sqrt{5} \approx 2.236 \quad \text{(using a calculator or square root table)}
\]
\[
5\sqrt{5} \approx 5 \times 2.236 \approx 11.18
\]

### Step 4: Calculate the perimeter of the triangle

The perimeter \(P\) is the sum of the lengths of the sides:

\[
P = d_{12} + d_{23} + d_{31} = 5 + 10 + 5\sqrt{5}
\]

Substituting the approximate value:

\[
P \approx 5 + 10 + 11.18 = 26.18
\]

### Step 5: Round the perimeter to the nearest tenth

Rounding \(26.18\) to the nearest tenth gives:

\[
P \approx 26.2
\]

Thus, the perimeter of the triangle, rounded to the nearest tenth of a unit, is:

\[
\boxed{26.2}
\]