A) Two chords, AB and CD of a circle whose radius is 13 cm are equal and parallel. If each chord is 12 cm long, find the distance between them. B) The line AB passes to points A(0,... A) Two chords, AB and CD of a circle whose radius is 13 cm are equal and parallel. If each chord is 12 cm long, find the distance between them. B) The line AB passes to points A(0,-5) and B(0,8). Use the line AB to answer the following questions: (i) Find the reflection matrix which defines the reflection through the line AB. (ii) Use the reflection matrix obtained from (i) to find the image of P(-5, -3) under reflection through that line AB.
Understand the Problem
The question consists of two parts: (a) involves finding the distance between two equal and parallel chords in a circle given the radius and chord length, requiring geometric understanding and application of the Pythagorean theorem. (b) concerns linear algebra and transformations, specifically finding a reflection matrix for a line defined by two points and then using that matrix to find the image of a point under reflection through the same line.
Answer
(a) $2\sqrt{133}$ cm (b) (i) $ \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} $ (ii) $(5, -3)$
Answer for screen readers
(a) The distance between the chords is $2\sqrt{133}$ cm.
(b) (i) The reflection matrix is $$ \begin{bmatrix} -1 & 0 \ 0 & 1 \end{bmatrix} $$ (ii) The image of $P(-5, -3)$ is $(5, -3)$.
Steps to Solve
- Find the distance from the center to one chord
Imagine a perpendicular line from the center of the circle to chord AB. This bisects the chord. We have a right triangle with hypotenuse 13 (radius) and one leg 6 (half of the chord length). Use the Pythagorean theorem to find the other leg (distance from center to chord):
$a^2 + b^2 = c^2$ $6^2 + b^2 = 13^2$ $36 + b^2 = 169$ $b^2 = 133$ $b = \sqrt{133}$
- Calculate the total distance between the chords
Since the chords are equal and parallel, they are equidistant from the center. There are two possibilities: the chords are on the same side of the center or on opposite sides. Since the question does not provide extra info, we should assume that two chords are on opposite sides. We must multiply the distance by 2
Distance = $2 \times \sqrt{133} = 2\sqrt{133}$
- Find the slope of the line AB
The line passes through $A(0, -5)$ and $B(0, 8)$. The slope $m$ is given by:
$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{8 - (-5)}{0 - 0} = \frac{13}{0}$
The slope is undefined, which means the line is vertical
- Understand reflection through a vertical line
Since the line AB is vertical and passes through x=0, it is the y axis.
Reflecting a point $(x, y)$ through the y-axis results in the point $(-x, y)$. Therefore, the reflection matrix is:
$$ \begin{bmatrix} -1 & 0 \ 0 & 1 \end{bmatrix} $$
- Find the image of P(-5, -3) under reflection
Multiply the reflection matrix by the coordinate matrix of point P(-5, -3):
$$ \begin{bmatrix} -1 & 0 \ 0 & 1 \end{bmatrix} \begin{bmatrix} -5 \ -3 \end{bmatrix}
\begin{bmatrix} (-1)(-5) + (0)(-3) \ (0)(-5) + (1)(-3) \end{bmatrix}
\begin{bmatrix} 5 \ -3 \end{bmatrix} $$
So, the image of $P(-5, -3)$ is $(5, -3)$.
(a) The distance between the chords is $2\sqrt{133}$ cm.
(b) (i) The reflection matrix is $$ \begin{bmatrix} -1 & 0 \ 0 & 1 \end{bmatrix} $$ (ii) The image of $P(-5, -3)$ is $(5, -3)$.
More Information
The key to part (a) is recognizing the geometric setup and using the Pythagorean theorem. For part (b), understanding how reflection matrices work, especially for simple cases like reflection over the y-axis, is crucial.
Tips
A common mistake in part (a) is only calculating the distance from the center to one chord and forgetting to double it, as the chords are on opposite sides of the center. In part (b), mistakes can arise from incorrectly calculating the slope or not recognizing that the line is simply the y-axis, which simplifies the reflection matrix. Another frequent mistake is confusing x and y coordinates when computing the reflection.
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