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Svara på alla matte frågor oavsett grafer trianglar och vad som helst. Du måste svara på det sättet som sägs. Om det behövs grafik lösning då måste du rita grafer också. Du behöver inte ge några förklaringar på frågor som säger”krävs endast svar”. Och måste läsa noga på alla frågor innan du svarar. 

Let's solve the given problems step by step.

### Problem 12
The figure shows a circle with two intersecting chords. We need to find the value of \( x \).

Using the intersecting chords theorem, which states:
\[
a \times b = c \times d
\]
where \( a \) and \( b \) are segments of one chord and \( c \) and \( d \) are segments of the other chord.

For the chords given:
- One chord is divided into segments of length \( 12 \) and \( x \).
- The other chord is divided into segments of \( 4x \) and \( 8 \).

Applying the intersecting chord theorem:
\[
12 \times x = 4x \times 8
\]

Solving this equation:
\[
12x = 32x
\]
\[
32x - 12x = 0
\]
\[
20x = 0
\]
\[
x = 0
\]

The answer is:
\[
x = 0
\]

### Problem 13
We have a triangle \( \triangle ABC \) with vertices:
- \( A = (1, 2) \)
- \( B = (3, -2) \)
- \( C = (5, 4) \)

We need to show that the height from point \( A \) to the base \( BC \) is exactly \( \sqrt{10} \) units.

To find the height from point \( A \) to the line \( BC \):
1. **Find the equation of line \( BC \)**:

   First, determine the slope \( m \) of line \( BC \):
   \[
   m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - (-2)}{5 - 3} = \frac{6}{2} = 3
   \]

   Now, use the point-slope form of a line with point \( B(3, -2) \):
   \[
   y - (-2) = 3(x - 3)
   \]
   \[
   y + 2 = 3x - 9
   \]
   \[
   y = 3x - 11
   \]

2. **Find the perpendicular distance from point \( A(1, 2) \) to the line \( BC \)**:

   The formula for the distance \( d \) from a point \( (x_1, y_1) \) to a line \( ax + by + c = 0 \) is:
   \[
   d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}}
   \]

   Rewrite the line equation \( y = 3x - 11 \) in standard form:
   \[
   3x - y - 11 = 0
   \]

   Plugging in \( A(1, 2) \):
   \[
   a = 3, \quad b = -1, \quad c = -11
   \]
   \[
   d = \frac{|3(1) + (-1)(2) - 11|}{\sqrt{3^2 + (-1)^2}}
   \]
   \[
   d = \frac{|3 - 2 - 11|}{\sqrt{9 + 1}}
   \]
   \[
   d = \frac{|-10|}{\sqrt{10}}
   \]
   \[
   d = \frac{10}{\sqrt{10}} = \sqrt{10}
   \]

The answer is:
\[
\sqrt{10}
\]

Do you have any questions or would you like further details on any part of the solution?

### 5 Related Questions:
1. How do you use the intersecting chords theorem for finding unknown lengths in a circle?
2. What are some alternative methods for determining the height of a triangle given three vertices?
3. How can you convert a line equation from point-slope form to standard form?
4. What is the significance of using the perpendicular distance formula in coordinate geometry?
5. How can the height from a vertex to the opposite side help in calculating the area of a triangle?

### Tip:
Remember that when dealing with circles, properties like the intersecting chords theorem, secant-tangent theorem, and properties of inscribed angles are extremely helpful in solving problems involving circle geometry.

I figuren nedan visas en cirkel med två kordor. Bestäm värdet på x. Triangeln ABC har tre hörn A = (1, 2), B = (3, -2) och C = (5, 4). Visa att höjden mot basen BC i triangeln är exakt sqrt(10) längdenheter.

Learn how to solve for the unknown variable x using the intersecting chords theorem in a circle, and find the height from vertex A to base BC in triangle ABC. This problem includes concepts of circle geometry, distance calculations, and triangle heights. Suitable for Grades 10-12.

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