Question

The triangle \(P Q R\) is inscribed in the circle \(x^{2}+y^{2}=25 .\) If \(Q\) and \(R\) have coordinates \((3,4)\) and \((-4,3)\) respectively, then \(\angle Q P R\) is equal to (a) \((\pi / 2)\) (b) \((\pi / 3)\) (c) \((\pi / 4)\) (d) \((\pi / 6)\)

Short answer

Answer: \(\angle QPR = \frac{\pi}{2}\).

Step-by-step solution

Calculate the vectors PQ and PR

Let Q = (3,4) and R = (-4,3), and let P = (x,y), then vectors PQ and PR can be represented as: PQ = P - Q = (x-3,y-4) PR = P - R = (x+4,y-3)

Calculate the dot product of PQ and PR

The dot product of PQ and PR can be given as: PQ ⋅ PR = (x-3)(x+4) + (y-4)(y-3)

Use dot product formula to calculate angle QPR

The dot product formula is: PQ ⋅ PR = |PQ| * |PR| * cos(∠QPR) We know that since P is on the circle x²+y²=25, |PQ|^2 = (x-3)²+(y-4)² = x²+y²-6x-8y+25 = 25-6x-8y |PR|^2 = (x+4)²+(y-3)² = x²+y²+8x-6y+25 = 25+8x-6y Now, let's plug these values into the dot product formula: (x-3)(x+4) + (y-4)(y-3) = [(25-6x-8y) * (25+8x-6y)]^(1/2) * cos(∠QPR)

Check which option is correct

Now we try out every option and see which one satisfies the above equation: (a) ∠QPR = π/2: In this case, cos(∠QPR) = 0, so the left-hand side of the equation becomes: (x-3)(x+4) + (y-4)(y-3) = 0 Simplify this equation, we get x²+y²-25 = 0, which is true as P lies on the circle x²+y²=25. As this satisfies the given equation, the answer is:

Angle QPR

:\(\angle QPR = \frac{\pi}{2}\).

Key concepts

Triangle Geometry

In triangle geometry, understanding the properties of triangles and knowing how to apply them is fundamental. A triangle consists of three sides, three angles, and three vertices.
In a circle, which is a special scenario, this triangle is often referred to as an *inscribed* triangle or a *cyclic* triangle, meaning all its vertices lie on the circle.

• An inscribed triangle has some interesting properties, such as the fact that the angle across the diameter of the circle is a right angle.
• Angles in inscribed triangles can help us solve problems by leveraging these unique geometric relationships, allowing for the calculation of unknown angles or side lengths.

Understanding these properties is crucial for mastering circle-related triangular problems.

Dot Product

The dot product is a crucial concept in vector mathematics. It helps you find the angle between two vectors. For vectors \( \mathbf{a} = (a_1, a_2) \) and \( \mathbf{b} = (b_1, b_2) \), their dot product is given by:
\[ \mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 \]

• This dot product also equals the product of the magnitudes of the vectors and the cosine of the angle between them: \( |\mathbf{a}| |\mathbf{b}| \cos(\theta) \).
• If the dot product is zero, the vectors are perpendicular.

Using the dot product formula, you can easily determine the angle between the vectors originating from specific points, like those given in coordinate geometry.

Coordinate Geometry

Coordinate geometry, or analytic geometry, combines algebra with geometry by using graphs and coordinates. This method provides a powerful tool for describing geometric figures algebraically and solving geometric problems.

• The basics involve finding distances, midpoints, and slopes of lines defined by coordinates.
• Equations of shapes like circles or parabolas can also be solved using coordinate principles. For example, the equation \( x^2 + y^2 = 25 \) signifies a circle centered at the origin with a radius of 5.

Coordinate geometry allows you to position points, lines, and shapes efficiently within the coordinate plane, making calculations simpler and more intuitive.

Trigonometry

Trigonometry deals with the relationships between the angles and sides of triangles, especially right triangles. In problems involving circles and inscribed angles, trigonometric principles are often applied to find the needed angles.

• The cosine function is particularly useful here, as it relates the angle to the dot product of vectors, helping find angles between intersecting lines or vectors.
• Common angles like \( \frac{\pi}{2} \), \( \frac{\pi}{3} \), \( \frac{\pi}{4} \), and \( \frac{\pi}{6} \) are often used and recognized in integer relations.

With trigonometry, we solve for unknown angles by leveraging these relationships, making sense of the geometric configurations and vector orientations.