Example

Problem: The diagram shows a wedge of Cheddar cheese, where rectangle PQRS is perpendicular (at 90°) to rectangle RSTU. The distances are PQ = 2.5m, QR = 7.8 m, and TU = 4.9 m. You need to calculate:

• (a) The distance $QT$
• (b) The angle $QTR$

Step-by-Step Solution:

1. Understanding the Problem:

• You need to find the distance $QT$, which is a line running through the $3D$ shape, and the angle $QTR$.
• To do this, you'll need to identify right-angled triangles within the shape.

2. Finding $QT$:

• Step 1: Spot the right-angled triangle involving $QT$. In this case, it's triangle QTR.
• Step 2: Break down the problem into two triangles:
• First, calculate $TR$ using the triangle $TRU$.
• Then use the triangle $QTR$ to find $QT$.

3. Calculating $TR$:

• Given: $TR$ is the hypotenuse of triangle $TRU$.

• Use Pythagoras' Theorem:

4. Calculating $QT$:

• $QT$ is the hypotenuse of the triangle $QTR$.
• Use the lengths $QR$ and $TR$:

5. Finding Angle $QTR$:

• Using Tan:

Final Answers:

• Distance $QT$: Approximately 9.54 m.
• Angle $QTR$: Approximately 15.2°.

Example Problem: A tent is shown in the diagram with a vertical pole OP. The pole is at the centre of rectangle $QRST$. Given that TQ=13 m, TS=5 m, and SR=12 m, find the height of the pole $OP$.

Step-by-Step Solution:

6. Breaking Down the Problem:

• Objective: Calculate the height $OP$ of the vertical pole.
• Identify relevant triangles and determine the lengths or angles that can be used to find $OP$.

7. Finding the Length $OT$:

• $OT$ is half the length of $TR$, since $O$ is the midpoint of $TR$.
• Step 1: Calculate $TR$ using Pythagoras' Theorem in the right-angled triangle $TRS$

Calculation:

• Therefore, $OT=\frac{TR}{2}=\frac{13}2=:highlight[6.5m]$.

1. Calculating the Height $OP$

• Now, consider the right-angled triangle $OTP$, where OT=6.5 m and ∠OTP=48°.
• Use the tangent ratio to find $OP$. Calculation: