To find \( x \) in the isosceles triangle \( \triangle ABC \) where \( AB = BC = 12 \) and \( BH \) is the altitude, we can start by identifying the structure of the triangle.
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Since \( H \) is the foot of the altitude from \( B \) to side \( AC \), it divides \( AC \) into two equal parts because \( \triangle ABC \) is isosceles. Let \( AH = CH = \frac{AC}{2} \).
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Let \( AC = 2y \), so \( AH = CH = y \).
Using the Pythagorean theorem in triangle \( \triangle ABH \):
\[ AB^2 = AH^2 + BH^2 \]
In our case, the values will be:
- \( AB = 12 \)
- \( AH = y \)
- \( BH = x \)
Plugging these into the equation, we have:
\[ 12^2 = y^2 + x^2 \] \[ 144 = y^2 + x^2 \tag{1} \]
Now, using triangle \( \triangle BHC \):
Using the Pythagorean theorem again:
\[ BC^2 = CH^2 + BH^2 \]
This is identical to our previous calculation as well because \( BC = AB \) and \( CH = AH \):
\[ 12^2 = y^2 + x^2 \] \[ 144 = y^2 + x^2 \tag{2} \]
Additionally, note that \( y \) represents half the base \( AC \).
Since we know \( y = \frac{AC}{2} \) and we want to apply the altitude \( BH \):
To find \( x \) in terms of \( y \), the in-depth triangle \( \triangle ABH \):
By substituting \( y \) with \( 12^2 - x^2 \):
So we need one more equation to solve this. Since \( BH \) divides the triangle into two right triangles, we can determine \( AC \).
To get \( AC \):
Using the altitude property \( x \) and the length from A or C.
Thus, we can find \( x \) and \( (y) \) as:
\[ y = \sqrt{12^2 - x^2} \]
But we conveniently find from \( \triangle ABH \): \[ y^2 + x^2 = 144 \]
Thus: \[ x^2 = 144 - y^2 \tag{3} \]
The approach verifies that we correctly represent the dimensions of the triangle.
Final Application: The Pythagorean theorem state in the specific configuration \( AB^2 = AH^2 + BH^2 \) on triangle \( ABH \).
Thus, the equation we want with \( x \):
\[ 12^2 = AH^2 + BH^2 = y^2 + x^2 \]
\[ \rightarrow 144 = y^2 + x^2 \]
Therefore, equation 1 can yield: Clearly leads us to formulating to apply \( x \) when substituting through both triangle relations representing:
- BH = x,
- AH = y,
- BC = 12 and
- Vertical altitude perpendicular from point to base proportional \( y = \sqrt{12^2 -x^2} \)'s properties.
Correlate forming \( AB^2 = AC^2 \) as measures retain defined outputs of right relations for confirmations.
Conclusively: The equation: \( 144 = y^2 + x^2 \) showcases the relation confirmed, specifically to develop \( x \) outputs through structural validations surrounding the Pythagorean theorem application definitively relevant to triangles.
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