Question

Explain how you could find the telemetered location of \(B\) if you know the locations of \(A\) and \(C\) and their internal angles.

Short answer

To find the telemetered location of \(B\), first calculate the third internal angle at point \(B\) using the Angle Sum Property of a Triangle. Then, apply the sine rule in triangle \(ABC\) to find the position of \(B\) relative to \(A\) and \(C\). By knowing the positions of \(A\) and \(C\), determine the telemetered location of \(B\).

Step-by-step solution

Understand the given situation and sketch a diagram

Identify and annotate the points \(A\), \(B\), and \(C\) in a diagram. Associate the given internal angles with their corresponding points on the sketch. Draw lines connecting these points and make sure they form a triangle.

Apply the Angle Sum Property of a Triangle

Since the angle sum property of a triangle states that the sum of the interior angles of a triangle is 180 degrees, calculate the third internal angle (at point \(B\)) by subtracting the sum of the two given internal angles from 180.

Use Trigonometric Ratios

Assuming that the distances between \(A\) and \(C\), and \(A\) and \(B\) are known (perhaps measured or given), apply sine rule in triangle \(ABC\) to conclude that \(\frac{BC}{sin∠A} = \frac{AB}{sin∠C}\). Solve this equation for \(BC\), which gives the location of point \(B\) relative to points \(A\) and \(C\).

Find the Telemetered Location

Knowledge about the locations of \(A\) and \(C\) combined with the calculated value of \(BC\), would enable you to determine the telemetered location of \(B\). If the coordinates of \(A\) and \(C\) are known, apply trigonometric relations to find the coordinates of \(B\) or use geometrical tools if the situation is represented on a map.

Key concepts

Triangle Interior Angles

Every triangle has three interior angles, which are the angles inside the triangle where the sides meet. One fascinating property of triangles is that the sum of their interior angles is always 180 degrees. Whether it's a small or large triangle, this rule never changes. This is known as the angle sum property of a triangle.

Let's say you're given two angles in a triangle, such as angles at points \(A\) and \(C\). You can easily find the third angle, \(\angle B\), by subtracting the sum of the two known angles from 180 degrees. This makes it incredibly handy in solving triangle problems because if you know two angles, you automatically know the third one too.

Sine Rule

The Sine Rule is a powerful tool in trigonometry used to solve problems in any triangle, not just right-angled ones. It involves the relationship between the sides of a triangle and its angles. Let’s explore what the Sine Rule is all about:

In any triangle, \(\triangle ABC\), the Sine Rule states that:

• \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)

Here, \(a, b,\) and \(c\) are the lengths of the sides opposite the angles \(A, B,\) and \(C\) respectively. This rule can help us find unknown side lengths or angles if we have some known values. For example, if we know sides \(AB\) and \(AC\) and angle \(\angle A\), we can use the Sine Rule to find side \(BC\), which in turn helps us to telemeter the position of point \(B\) based on the given data about points \(A\) and \(C\).

Geometric Diagram

A geometric diagram is a visual representation that helps in understanding and solving mathematical problems. In our case of locating point \(B\), a diagram helps to clearly lay out the triangle \(\triangle ABC\), establishing relationships between the points and angles.

Start by marking points \(A\), \(B\), and \(C\) on your paper. Connect these points to form the triangle. Label the given angles and sides, if any.

• Drawing an accurate diagram provides a visual cue, ensuring clarity in analyzing the problem.
• It also aids in correctly applying formulas such as the Sine Rule, as it encapsulates an overview of the triangle’s structure.

By setting out a geometric diagram at the start, troubleshooting becomes easier, and mathematical concepts become much more comprehensible.

Coordinate Geometry

Coordinate geometry merges algebra and geometry, providing a powerful framework for solving geometric problems using a coordinate plane. When working with the locations of points \(A\) and \(C\), you may be given their coordinates. Using coordinate geometry can thus simplify the task of finding point \(B\).

By knowing the coordinates of points \(A\) and \(C\), alongside the distances measured or calculated using trigonometric formulas like the Sine Rule, you can find the coordinates for point \(B\). Here's how:

• Define a coordinate system placing \(A\) and \(C\), usually along a known axis for simplification.
• Use distance formulas and trigonometry to compute the exact location of \(B\) relative to these points.
• This process integrates numerical methods with graphical visualization, balancing conceptual understanding with mathematical verification.

By representing geometric figures in a coordinate plane, complex calculations become straightforward, providing a systematic approach to determining the precise location of points.