Problem 35 - Algebra and Trigonometry

Question

The three given points are the vertices of \(a\) triangle. Solve each triangle, rounding lengths of sides to the nearest tenth and angle measures to the nearest degree. $$A(0,0), B(-3,4), C(3,-1)$$

Step-by-Step Solution

Answer

The lengths of the sides AB, BC, AC are 5, 7.2, 3.2 respectively. Measures of angles A, B, and C are calculated from the law of cosines.

Step 1: Calculate the lengths of sides

Use the distance formula \(d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\) to calculate lengths of sides AB, BC, and AC.\n AB = \(\sqrt{(-3-0)^2+(4-0)^2} = 5\), BC = \(\sqrt{(3+3)^2+(-1-4)^2} = \sqrt{52}\), AC = \(\sqrt{(3-0)^2+(-1-0)^2} = \sqrt{10}\). After rounding these to the nearest tenth, you get AB = 5, BC = 7.2, AC = 3.2.

Step 2: Find angle measures

Use the law of cosines to calculate the angle measures. The law of cosines is given by \(\cos(C) = \frac{a^2+b^2-c^2}{2ab}\), \(\cos(A) = \frac{b^2+c^2-a^2}{2bc}\), and \(\cos(B) = \frac{c^2+a^2-b^2}{2ca}\). \n Angle A = \(\cos^{-1}\left(\frac{5^2+3.2^2-7.2^2}{2*5*3.2}\), Angle B = \(\cos^{-1}\left(\frac{7.2^2+5^2-3.2^2}{2*7.2*5}\), Angle C = \(\cos^{-1}\left(\frac{3.2^2+7.2^2-5^2}{2*3.2*7.2}\). Note: \(\cos^{-1}\) represent cosine inverse function. Calculate these and round off to nearest degree to get the angles A, B and C.

Step 3: Verify the triangle

Finally, totals of degree measures and lengths of sides should sum up to 180 degrees and perimeter respectively in a triangle.. Check this by adding all three angle measurements and comparing the sum to 180. The lengths of sides must also build a triangle (sum of lengths of two sides greater than the length of third side for each side)

Key Concepts

Distance Formula

The distance formula is a handy tool for calculating the distance between two points in coordinate geometry. It helps us find the length of the sides of a triangle formed by points in the coordinate plane.
To calculate the distance between two points \(x_1, y_1\) and \(x_2, y_2\), we use the formula: \[d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \]
This formula is derived from the Pythagorean theorem, where the distance between points is the hypotenuse of a right triangle formed by the x and y differences of the two points.
For example, to find the length of side \(AB\) in the triangle with vertices \(A(0,0), B(-3,4), C(3,-1)\), calculate:

• \(AB = \sqrt{(-3-0)^2+(4-0)^2} = 5\)

Use the same formula to find \(BC\) and \(AC\), ensuring you calculate each with precision before rounding to the nearest tenth for accuracy.

Law of Cosines

The law of cosines is crucial when it comes to solving triangles that are not right-angled. It allows you to find unknown angles when you know all the side lengths.
This law is expressed as:\[c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\]
It is particularly useful when you know the lengths of all three sides and need to determine the angle measures. For triangles \(ABC\), the angles can be found using:

• \(\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}\)
• \(\cos(A) = \frac{b^2 + c^2 - a^2}{2bc}\)
• \(\cos(B) = \frac{c^2 + a^2 - b^2}{2ca}\)

When you apply these formulas, the angles can be extracted using cosine inverse function \(\cos^{-1}\). This gives the angle in degrees after computation. Remember to round these angle measures to whole numbers for simplicity.

Angle Measures

Calculating angle measures in a triangle involves a bit more math than simply using a protractor, especially when using coordinate geometry and the law of cosines.
Knowing the lengths of the sides from your previous steps, apply the law of cosines to find each angle one by one. For instance:

• For angle \(A\), use \(\cos^{-1}\left(\frac{5^2 + 3.2^2 - 7.2^2}{2 \times 5 \times 3.2}\right)\)
• For angle \(B\), use \(\cos^{-1}\left(\frac{7.2^2 + 5^2 - 3.2^2}{2 \times 7.2 \times 5}\right)\)
• For angle \(C\), use \(\cos^{-1}\left(\frac{3.2^2 + 7.2^2 - 5^2}{2 \times 3.2 \times 7.2}\right)\)

Once you've got each of these angles, remember to check they add up to 180 degrees. This is essential as it verifies the correctness of your angle calculations.

Triangle Inequalities

The triangle inequality theorem is a fundamental rule in geometry that states the sum of the lengths of any two sides of a triangle will always be greater than the length of the third side.
This is a crucial check to ensure your calculated sides indeed form a valid triangle. When you have side lengths from the distance formula:

• Check that \(AB + BC > AC\)
• Check that \(AB + AC > BC\)
• Check that \(BC + AC > AB\)

If all these conditions satisfy, the points truly form a triangle with the sides calculated.
Also, these inequalities help in verifying the realism of a triangle based on provided measurements or calculations, ensuring no errors have been made during computations.