Distance and Midpoint Formulas – Precalculus Study Guidance

Q1. Find the distance between the points (1, 3) and (2, 5).

Background

Topic: Distance Formula

This question tests your ability to calculate the distance between two points in the coordinate plane using the distance formula.

Key formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Where:

• $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the two points.

• $d$ is the distance between the points.

Step-by-Step Guidance

1. Identify the coordinates: $(x_1, y_1) = (1, 3)$ and $(x_2, y_2) = (2, 5)$.

2. Subtract the $x$-coordinates: $x_2 - x_1 = 2 - 1$.

3. Subtract the $y$-coordinates: $y_2 - y_1 = 5 - 3$.

4. Square each difference: $(x_2 - x_1)^2$ and $(y_2 - y_1)^2$.

5. Add the squared differences and set up the square root: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Try solving on your own before revealing the answer!

Final Answer: $\sqrt{5}$

$d = \sqrt{(2-1)^2 + (5-3)^2} = \sqrt{1^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5}$

The exact distance between the points is $\sqrt{5}$ units.

Q2. Find the distance between the points (−1, 6) and (−11, 2).

Background

Topic: Distance Formula

This question asks you to use the distance formula to find the length between two points with negative coordinates.

Key formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Step-by-Step Guidance

1. Identify the coordinates: $(x_1, y_1) = (-1, 6)$ and $(x_2, y_2) = (-11, 2)$.

2. Subtract the $x$-coordinates: $x_2 - x_1 = -11 - (-1)$.

3. Subtract the $y$-coordinates: $y_2 - y_1 = 2 - 6$.

4. Square each difference: $(x_2 - x_1)^2$ and $(y_2 - y_1)^2$.

5. Add the squared differences and set up the square root: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Try solving on your own before revealing the answer!

Final Answer: $\sqrt{104}$

$d = \sqrt{(-11 - (-1))^2 + (2 - 6)^2} = \sqrt{(-10)^2 + (-4)^2} = \sqrt{100 + 16} = \sqrt{116}$

After simplifying, $\sqrt{116} = 2\sqrt{29}$, so the exact distance is $2\sqrt{29}$ units.

Q3. Find the distance between the points (12, 5) and (12, 3/2).

Background

Topic: Distance Formula

This question involves finding the distance between two points that share the same $x$-coordinate, which means the segment is vertical.

Key formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Step-by-Step Guidance

1. Identify the coordinates: $(x_1, y_1) = (12, 5)$ and $(x_2, y_2) = (12, \frac{3}{2})$.

2. Subtract the $x$-coordinates: $x_2 - x_1 = 12 - 12 = 0$.

3. Subtract the $y$-coordinates: $y_2 - y_1 = \frac{3}{2} - 5$.

4. Square each difference: $(x_2 - x_1)^2$ and $(y_2 - y_1)^2$.

5. Add the squared differences and set up the square root: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Try solving on your own before revealing the answer!

Final Answer: $\frac{7}{2}$

$d = \sqrt{(12-12)^2 + (\frac{3}{2} - 5)^2} = \sqrt{0 + (-\frac{7}{2})^2} = \frac{7}{2}$

The distance is $\frac{7}{2}$ units, since the points are vertically aligned.

Q4. What is the length of the hypotenuse of the triangle below?

Background

Topic: Pythagorean Theorem

This question tests your ability to use the Pythagorean Theorem to find the length of the hypotenuse in a right triangle.

Key formula:

$c = \sqrt{a^2 + b^2}$

Where:

• $a$ and $b$ are the lengths of the legs of the triangle.

• $c$ is the length of the hypotenuse.

Step-by-Step Guidance

1. Identify the lengths of the two legs, $a$ and $b$ (refer to the diagram or given values).

2. Square each leg: $a^2$ and $b^2$.

3. Add the squares: $a^2 + b^2$.

4. Take the square root to find $c$: $c = \sqrt{a^2 + b^2}$.

Try solving on your own before revealing the answer!

Final Answer: (Depends on triangle values)

Since the actual leg lengths are not specified in the worksheet excerpt, please refer to your diagram and use $c = \sqrt{a^2 + b^2}$ with your values.

Q5. Find the coordinates of the midpoint of line segment AB for the points A = (1, 2) and B = (11, 3).

Background

Topic: Midpoint Formula

This question tests your ability to find the midpoint between two points in the coordinate plane.

Key formula:

$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

Where:

• $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the endpoints.

• $M$ is the midpoint.

Step-by-Step Guidance

1. Identify the coordinates: $A = (1, 2)$ and $B = (11, 3)$.

2. Add the $x$-coordinates: $1 + 11$.

3. Add the $y$-coordinates: $2 + 3$.

4. Divide each sum by 2 to find the midpoint: $M = \left( \frac{1 + 11}{2}, \frac{2 + 3}{2} \right)$.

Try solving on your own before revealing the answer!

Final Answer: $(6, 2.5)$

$M = \left( \frac{1 + 11}{2}, \frac{2 + 3}{2} \right) = (6, 2.5)$

The midpoint is at $(6, 2.5)$.

Q6. Find the coordinates of the midpoint of line segment AB for the points A = (−5, 2) and B = (−4, −1).

Background

Topic: Midpoint Formula

This question tests your ability to find the midpoint between two points with negative coordinates.

Key formula:

$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

Step-by-Step Guidance

1. Identify the coordinates: $A = (-5, 2)$ and $B = (-4, -1)$.

2. Add the $x$-coordinates: $-5 + (-4)$.

3. Add the $y$-coordinates: $2 + (-1)$.

4. Divide each sum by 2 to find the midpoint: $M = \left( \frac{-5 + (-4)}{2}, \frac{2 + (-1)}{2} \right)$.

Try solving on your own before revealing the answer!

Final Answer: $(-4.5, 0.5)$

$M = \left( \frac{-5 + (-4)}{2}, \frac{2 + (-1)}{2} \right) = (-4.5, 0.5)$

The midpoint is at $(-4.5, 0.5)$.

Q7. Challenging: A line segment AB has midpoint M = (2, 5). If A = (1, 1), what are the coordinates of point B?

Background

Topic: Midpoint Formula (Solving for an endpoint)

This question tests your ability to use the midpoint formula in reverse to find the missing endpoint when given one endpoint and the midpoint.

Key formula:

$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

Step-by-Step Guidance

1. Let $A = (x_1, y_1) = (1, 1)$ and $M = (2, 5)$.

2. Let $B = (x_2, y_2)$ be the unknown endpoint.

3. Set up the midpoint equations: $2 = \frac{1 + x_2}{2}$ and $5 = \frac{1 + y_2}{2}$.

4. Solve each equation for $x_2$ and $y_2$ by multiplying both sides by 2 and then subtracting 1.

Try solving on your own before revealing the answer!

Final Answer: $B = (3, 9)$

Solving: $2 = \frac{1 + x_2}{2} \Rightarrow x_2 = 3$ and $5 = \frac{1 + y_2}{2} \Rightarrow y_2 = 9$.

The coordinates of point $B$ are $(3, 9)$.