Certainly! Let's break down the solution step by step.
### Step 1: Understand the Geometry and Missing Lengths
We start with the given right-angled triangle [tex]\( \triangle GAP \)[/tex]:
- [tex]\( AP = 12 \, m \)[/tex]
- [tex]\( GA = 5 \, m \)[/tex]
Since [tex]\( \triangle GAP \)[/tex] is a right-angled triangle with [tex]\( \angle GAP = 90^\circ \)[/tex], we can use the Pythagorean theorem to find [tex]\( GP \)[/tex]:
[tex]\[ GP = \sqrt{AP^2 - GA^2} \][/tex]
[tex]\[ GP = \sqrt{12^2 - 5^2} \][/tex]
[tex]\[ GP = \sqrt{144 - 25} \][/tex]
[tex]\[ GP = \sqrt{119} \][/tex]
### Step 2: Determine Similar Triangles and Lengths
The point [tex]\( T \)[/tex] on [tex]\( AP \)[/tex] is such that [tex]\( \angle AGT = \angle TGP = x \)[/tex]. This implies the triangles [tex]\( \triangle AGT \)[/tex] and [tex]\( \triangle TGP \)[/tex] are similar and isosceles at [tex]\( G \)[/tex], thus:
[tex]\[ AG = GT \][/tex]
Since [tex]\( AG = 5 \, m \)[/tex], it must be that [tex]\( GT = 5 \, m \)[/tex].
### Step 3: Find Length [tex]\( AT \)[/tex]
Knowing [tex]\( T \)[/tex] divides [tex]\( AP \)[/tex] such that [tex]\( AT + TP = AP \)[/tex], and [tex]\( TP = GT = 5 \, m \)[/tex]:
[tex]\[ AT = AP - GT \][/tex]
[tex]\[ AT = 12 \, m - 5 \, m \][/tex]
[tex]\[ AT = 7 \, m \][/tex]
### Step 4: Determine Tangent Values
Next, we need to find [tex]\( \tan x \)[/tex] in [tex]\( \triangle AGT \)[/tex]:
[tex]\[ \tan x = \frac{GA}{AT} \][/tex]
[tex]\[ \tan x = \frac{5}{7} \][/tex]
### Step 5: Using the Double Angle Formula for Tangent
Using the double angle formula for the tangent, we have:
[tex]\[ \tan (2x) = \frac{2 \tan x}{1 - \tan^2 x} \][/tex]
Substitute [tex]\( \tan x = \frac{5}{7} \)[/tex]:
[tex]\[ \tan^2 x = \left(\frac{5}{7}\right)^2 = \frac{25}{49} \][/tex]
[tex]\[ \tan (2x) = \frac{2 \cdot \frac{5}{7}}{1 - \frac{25}{49}} \][/tex]
[tex]\[ \tan (2x) = \frac{\frac{10}{7}}{\frac{24}{49}} \][/tex]
[tex]\[ \tan (2x) = \frac{10}{7} \times \frac{49}{24} \][/tex]
[tex]\[ \tan (2x) = \frac{10 \cdot 49}{7 \cdot 24} \][/tex]
[tex]\[ \tan (2x) = \frac{490}{168} = \frac{245}{84} = \frac{35}{12} \][/tex]
Thus, without using a calculator, we find:
### Final Answers
a) The exact value of [tex]\( \tan(2x) \)[/tex] is:
[tex]\[ \tan(2x) = \frac{35}{12} \][/tex]
b) The exact value of [tex]\( \tan(x) \)[/tex] by using the double angle formula is:
[tex]\[ \tan(x) = \frac{5}{7} \][/tex]
c) The exact value of [tex]\( AT \)[/tex] is:
[tex]\[ AT = 7 \, m \][/tex]
These are the exact values as derived geometrically and algebraically.
