Question

we do: example
the polynesian triangle formed between the three pacific island groups in the
south pacific region is known as polynesia.
the approximate coordinates in latitude and longitude of each vertex are
auckland, new zealand (-40.9, 174.9), honolulu, hawaii (21.3, -157.9), and easter
island (-27.1, -109.4).
use coordinate geometry to determine the type of triangle formed.

Explanation:

Step1: Recall the distance formula

The distance between two points \((x_1,y_1)\) and \((x_2,y_2)\) is given by \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)

Step2: Calculate distance between Auckland (A: \(- 40.9,174.9\)) and Honolulu (H: \(21.3,-157.9\))

\[ \begin{align*} d_{AH}&=\sqrt{(21.3 - (-40.9))^2+(-157.9 - 174.9)^2}\\ &=\sqrt{(62.2)^2+(-332.8)^2}\\ &=\sqrt{3868.84 + 110755.84}\\ &=\sqrt{114624.68} \end{align*} \]

Step3: Calculate distance between Auckland (A: \(-40.9,174.9\)) and Easter Island (E: \(-27.1,-109.4\))

\[ \begin{align*} d_{AE}&=\sqrt{(-27.1-(-40.9))^2+(-109.4 - 174.9)^2}\\ &=\sqrt{(13.8)^2+(-284.3)^2}\\ &=\sqrt{190.44+80826.49}\\ &=\sqrt{81016.93} \end{align*} \]

Step4: Calculate distance between Honolulu (H: \(21.3,-157.9\)) and Easter Island (E: \(-27.1,-109.4\))

\[ \begin{align*} d_{HE}&=\sqrt{(-27.1 - 21.3)^2+(-109.4-(-157.9))^2}\\ &=\sqrt{(-48.4)^2+(48.5)^2}\\ &=\sqrt{2342.56+2352.25}\\ &=\sqrt{4694.81} \end{align*} \]

Step5: Now, check the squares of the distances (to avoid square - root calculations for comparison)

\(d_{AH}^2=114624.68\), \(d_{AE}^2 = 81016.93\), \(d_{HE}^2=4694.81\)

We check the Pythagorean theorem: \(d_{HE}^2 + d_{AE}^2=4694.81 + 81016.93=85711.74 eq114624.68=d_{AH}^2\)

Check if two sides are equal: \(d_{AH}\approx338.56\), \(d_{AE}\approx284.63\), \(d_{HE}\approx68.52\). None of the distances are equal. Wait, let's re - calculate the distances more accurately.

Wait, let's recalculate \(d_{AH}\):

\(x\) - difference: \(21.3+40.9 = 62.2\), \(y\) - difference: \(- 157.9-174.9=-332.8\)

\(d_{AH}=\sqrt{62.2^{2}+(-332.8)^{2}}=\sqrt{3868.84 + 110755.84}=\sqrt{114624.68}\approx338.56\)

\(d_{AE}\): \(x\) - difference: \(-27.1 + 40.9 = 13.8\), \(y\) - difference: \(-109.4-174.9=-284.3\)

\(d_{AE}=\sqrt{13.8^{2}+(-284.3)^{2}}=\sqrt{190.44 + 80826.49}=\sqrt{81016.93}\approx284.63\)

\(d_{HE}\): \(x\) - difference: \(-27.1-21.3=-48.4\), \(y\) - difference: \(-109.4 + 157.9 = 48.5\)

\(d_{HE}=\sqrt{(-48.4)^{2}+48.5^{2}}=\sqrt{2342.56+2352.25}=\sqrt{4694.81}\approx68.52\)

Now, check the angles using the Law of Cosines. Let's check the angle at E.

\(\cos E=\frac{d_{HE}^2 + d_{AE}^2-d_{AH}^2}{2\times d_{HE}\times d_{AE}}\)

\(d_{HE}^2 + d_{AE}^2 - d_{AH}^2=4694.81+81016.93 - 114624.68=85711.74 - 114624.68=-28912.94\)

\(2\times d_{HE}\times d_{AE}=2\times68.52\times284.63\approx2\times19500\approx39000\)

\(\cos E=\frac{- 28912.94}{39000}\approx - 0.741\), so \(E\) is an obtuse angle.

Now, let's check if we made a mistake in coordinates. Wait, the longitude of Auckland: 174.9 degrees east, Honolulu: 157.9 degrees west, so the difference in longitude between Auckland and Honolulu is \(174.9+157.9 = 332.8\) (since they are on opposite sides of the prime meridian), which matches our \(y\) - difference (if we consider longitude as \(y\) - coordinate). The latitude difference between Auckland (\(-40.9\) south) and Honolulu (\(21.3\) north) is \(21.3 + 40.9=62.2\) (latitude as \(x\) - coordinate), which is correct.

For Easter Island: latitude \(-27.1\) (south), longitude \(-109.4\) (west). The difference in latitude between Auckland (\(-40.9\)) and Easter Island (\(-27.1\)) is \(-27.1+40.9 = 13.8\), difference in longitude: \(-109.4 - 174.9=-284.3\) (since Auckland is at \(174.9\) east and Easter Island at \(109.4\) west, so the difference is \(174.9 + 109.4=284.3\) west, so \(- 284.3\) in our coordinate system where east is positive? Wait, maybe we should consider longitude as a circular measure. But in the coordinate system given, we can proceed with the given coordinates.

Wait, let's recalculate the distances with correct sign handling.

For \(d_{AH}\):

\(x_1=-40.9,y_1 = 174.9\); \(x_2 = 21.3,y_2=-157.9\)

\(\Delta x=21.3-(-40.9)=62.2\), \(\Delta y=-157.9 - 174.9=-332.8\)

\(d_{AH}=\sqrt{(62.2)^{2}+(-332.8)^{2}}=\sqrt{3868.84 + 110755.84}=\sqrt{114624.68}\approx338.56\)

For \(d_{AE}\):

\(x_1=-40.9,y_1 = 174.9\); \(x_2=-27.1,y_2=-109.4\)

\(\Delta x=-27.1-(-40.9)=13.8\), \(\Delta y=-109.4 - 174.9=-284.3\)

\(d_{AE}=\sqrt{(13.8)^{2}+(-284.3)^{2}}=\sqrt{190.44+80826.49}=\sqrt{81016.93}\approx284.63\)

For \(d_{HE}\):

\(x_1 = 21.3,y_1=-157.9\); \(x_2=-27.1,y_2=-109.4\)

\(\Delta x=-27.1 - 21.3=-48.4\), \(\Delta y=-109.4-(-157.9)=48.5\)

\(d_{HE}=\sqrt{(-48.4)^{2}+(48.5)^{2}}=\sqrt{2342.56 + 2352.25}=\sqrt{4694.81}\approx68.52\)

Now, let's check the sum of squares of the two smaller sides: \(d_{HE}^2 + d_{AE}^2=4694.81+81016.93 = 85711.74\) and \(d_{AH}^2 = 114624.68\). Since \(d_{HE}^2 + d_{AE}^2<d_{AH}^2\), by the converse of the Pythagorean theorem, the triangle is obtuse. Also, since all three sides have different lengths (\(d_{AH}\approx338.56\), \(d_{AE}\approx284.63\), \(d_{HE}\approx68.52\)), the triangle is an obtuse scalene triangle.

Wait, but maybe we made a mistake in the coordinate interpretation. Let's consider that longitude is a circular measure, so the difference in longitude between two points \(l_1\) and \(l_2\) is \(\min(|l_1 - l_2|,360-|l_1 - l_2|)\). Let's recalculate the longitude differences correctly.

For Auckland (longitude \(174.9^{\circ}\) E) and Honolulu (longitude \(157.9^{\circ}\) W). The total longitude difference is \(174.9+157.9 = 332.8^{\circ}\), which is less than \(360^{\circ}\), so that's correct.

For Auckland ( \(174.9^{\circ}\) E) and Easter Island ( \(109.4^{\circ}\) W). Longitude difference: \(174.9 + 109.4=284.3^{\circ}\), correct.

For Honolulu ( \(157.9^{\circ}\) W) and Easter Island ( \(109.4^{\circ}\) W). Longitude difference: \(157.9 - 109.4 = 48.5^{\circ}\), correct. Latitude difference between Honolulu ( \(21.3^{\circ}\) N) and Easter Island ( \(27.1^{\circ}\) S) is \(21.3+27.1 = 48.4^{\circ}\), correct.

So our distance calculations are correct.

Now, let's check the squares again:

\(d_{HE}^2=( \text{latitude difference})^2+( \text{longitude difference})^2=(48.4)^2+(48.5)^2\approx2342.56 + 2352.25 = 4694.81\)

\(d_{AE}^2=(13.8)^2+(284.3)^2\approx190.44+80826.49 = 81016.93\)

\(d_{AH}^2=(62.2)^2+(332.8)^2\approx3868.84+110755.84 = 114624.68\)

Now, \(d_{HE}^2 + d_{AE}^2=4694.81 + 81016.93=85711.74\) and \(d_{AH}^2 = 114624.68\). Since \(85711.74<114624.68\), angle at E is obtuse. And since all sides are of different lengths, the triangle is an obtuse scalene triangle.

Answer:

The triangle formed is an obtuse scalene triangle.