Question

For the given points \(P, Q,\) and \(R,\) find the approximate measurements of the angles of \(\triangle P Q R\). $$P(1,-4), Q(2,7), R(-2,2)$$

Short answer

Answer: The approximate angles of the triangle are P ≈ 125.94°, Q ≈ 34.78°, and R ≈ 19.27°.

Step-by-step solution

Find the lengths of the sides of the triangle

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) to find the lengths of sides \(PQ\), \(QR\), and \(RP\). \(d_{PQ} = \sqrt{(2 - 1)^2 + (7 - (-4))^2} = \sqrt{1^2 + 11^2} = \sqrt{122}\) \(d_{QR} = \sqrt{(-2 - 2)^2 + (2 - 7)^2} = \sqrt{4^2 + (-5)^2} = \sqrt{41}\) \(d_{RP} = \sqrt{(-2 - 1)^2 + (2 - (-4))^2} = \sqrt{(-3)^2 + 6^2} = \sqrt{45}\)

Use the cosine rule to find the angles

Let the angles be \(P(\angle PRQ)\), \(Q(\angle QRP)\) and \(R(\angle RPQ)\). Now, we can use the cosine rule to find these angles: For angle P: \(\cos P = \frac{d_{QR}^2 + d_{RP}^2 - d_{PQ}^2}{2 d_{QR}d_{RP}} = \frac{41 + 45 - 122}{2\sqrt{41}\sqrt{45}} = -\frac{36}{\sqrt{41}\sqrt{45}}\) Now, calculate the angle: \(P = \arccos(-\frac{36}{\sqrt{41}\sqrt{45}}) \approx 125.94^\circ\) For angle Q: \(\cos Q = \frac{d_{RP}^2 + d_{PQ}^2 - d_{QR}^2}{2 d_{RP}d_{PQ}} = \frac{45 + 122 - 41}{2\sqrt{45}\sqrt{122}} = \frac{126}{\sqrt{45}\sqrt{122}}\) Now, calculate the angle: \(Q = \arccos(\frac{126}{\sqrt{45}\sqrt{122}}) \approx 34.78^\circ\) For angle R: \(\cos R = \frac{d_{PQ}^2 + d_{QR}^2 - d_{RP}^2}{2 d_{PQ}d_{QR}} = \frac{122 + 41 - 45}{2\sqrt{122}\sqrt{41}} = \frac{118}{\sqrt{122}\sqrt{41}}\) Now, calculate the angle: \(R = \arccos(\frac{118}{\sqrt{122}\sqrt{41}}) \approx 19.27^\circ\) Thus, the approximate measurements of the angles of triangle \(PQR\) are: \(P \approx 125.94^\circ\), \(Q \approx 34.78^\circ\), and \(R \approx 19.27^\circ\).

Key concepts

Understanding the Cosine Rule

The Cosine Rule is an essential tool in trigonometry that helps us find unknown angles or side lengths in any triangle. This rule becomes particularly useful in non-right-angled triangles, where traditional trigonometric ratios like sine, cosine, and tangent fall short.
To apply the Cosine Rule, we focus on the relationship between the lengths of the sides and a specific angle of the triangle.

• The formula is given by: a² = b² + c² - 2bc \cdot \cos(A)

Where:

• \(a\) is the side opposite angle \(A\)
• \(b\) and \(c\) are the other two sides of the triangle
• \(\cos(A)\) represents the cosine of the angle \(A\)

This rule allows us to solve for the angle \(A\) if we know the side lengths. By rearranging the equation for \(\cos(A)\), we calculate the angle using the arccosine function. This involves ensuring the relationships amongst the sides and angles are consistent with what a geometric triangle permits.
The ability to evaluate any angle given three sides makes the Cosine Rule a versatile method to analyze triangles fully.

Calculating Distance Between Points

The Distance Formula helps us find the straight-line distance between two points in a plane. This formula is rooted in the Pythagorean theorem and provides an effective way to deal with coordinate geometry.
The general formula for distance in a two-dimensional plane is:

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

This expression calculates the distance \(d\) between points \((x_1, y_1)\) and \((x_2, y_2)\).

• Begin by finding the difference between the x-coordinates: \((x_2 - x_1)\)
• Find the difference between the y-coordinates: \((y_2 - y_1)\)
• Square each of these differences to eliminate any negative results

Finally, add the squared differences and take the square root of the sum to compute the final distance.
Using this formula, distances such as \(d_{PQ}, d_{QR},\) and \(d_{RP}\) in triangles can be precisely calculated, which is crucial for applying the Cosine Rule effectively.

Angle Calculation in Triangles

Calculating angles within a triangle involves understanding both the geometric and trigonometric properties of the triangle itself. Once we compute the side lengths using the Distance Formula, we can move to angle calculation using the Cosine Rule.
In a triangle, the sum of all internal angles always equals \(180^\circ\). When sides \(a, b,\), and \(c\) are known, we can determine angles \(A, B,\), and \(C\) using:

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

Through these expressions, we derive \(\cos\) values that lead to calculating respective angles through the arccosine function.
This approach clarifies the triangle's angle measures, completing our comprehensive understanding of the triangle's properties and aiding in various practical applications, such as navigation and construction.