Precalculus Study Guide: Step-by-Step Guidance for Unit 1

Q1. Find the distance between the points P₁ = (4, 7) and P₂ = (−2, −3).

Background

Topic: Distance Formula in the Coordinate Plane

This question tests your ability to use the distance formula to find the length between two points in the Cartesian plane.

Key Terms and Formula:

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

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

Step-by-Step Guidance

1. Identify the coordinates: $P_1 = (4, 7)$ and $P_2 = (−2, −3)$.

2. Substitute the coordinates into the distance formula: $d = \sqrt{((-2) - 4)^2 + ((-3) - 7)^2}$.

3. Simplify the expressions inside the parentheses for both $x$ and $y$ differences.

4. Square each difference and add them together under the square root.

Try solving on your own before revealing the answer!

Final Answer: $d = \sqrt{36 + 100} = \sqrt{136} \approx 11.66$

We calculated the differences, squared them, added, and took the square root to find the distance.

Q2. Find the distance between P₁ = (−1, 3) and P₂ = (2, 4).

Background

Topic: Distance Formula in the Coordinate Plane

This question also tests your ability to use the distance formula to find the length between two points.

Key Terms and Formula:

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

Step-by-Step Guidance

1. Identify the coordinates: $P_1 = (−1, 3)$ and $P_2 = (2, 4)$.

2. Substitute into the formula: $d = \sqrt{(2 - (−1))^2 + (4 - 3)^2}$.

3. Simplify the differences inside the parentheses.

4. Square the results and add them under the square root.

Try solving on your own before revealing the answer!

Final Answer: $d = \sqrt{9 + 1} = \sqrt{10} \approx 3.16$

After substituting and simplifying, the distance is approximately 3.16 units.

Q3. Find the midpoint between P₁ = (4, 7) and P₂ = (−2, −3).

Background

Topic: Midpoint Formula

This question tests your ability to find the point exactly halfway between two given points in the plane.

Key Terms and Formula:

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

Step-by-Step Guidance

1. Identify the coordinates: $P_1 = (4, 7)$ and $P_2 = (−2, −3)$.

2. Add the $x$-coordinates together and divide by 2: $\frac{4 + (−2)}{2}$.

3. Add the $y$-coordinates together and divide by 2: $\frac{7 + (−3)}{2}$.

4. Write the midpoint as an ordered pair using your results from steps 2 and 3.

Try solving on your own before revealing the answer!

Final Answer: $M = (1, 2)$

The midpoint is found by averaging the $x$ and $y$ values of the two points.

Q4. Find the midpoint between P₁ = (−1, 3) and P₂ = (2, 4).

Background

Topic: Midpoint Formula

This question also tests your ability to find the midpoint between two points.

Key Terms and Formula:

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

Step-by-Step Guidance

1. Identify the coordinates: $P_1 = (−1, 3)$ and $P_2 = (2, 4)$.

2. Add the $x$-coordinates and divide by 2: $\frac{−1 + 2}{2}$.

3. Add the $y$-coordinates and divide by 2: $\frac{3 + 4}{2}$.

4. Write the midpoint as an ordered pair.

Try solving on your own before revealing the answer!

Final Answer: $M = (0.5, 3.5)$

The midpoint is halfway between the two points in both $x$ and $y$ directions.

Q5. Determine whether the points P₁ = (−5, 3), P₂ = (6, 0), and P₃ = (5, 5) form a right triangle.

Background

Topic: Distance Formula and Right Triangles

This question tests your ability to use the distance formula to find the lengths of the sides of a triangle and then determine if the triangle is a right triangle using the Pythagorean Theorem.

Key Terms and Formulas:

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

• Pythagorean Theorem: For a right triangle, $a^2 + b^2 = c^2$ where $c$ is the hypotenuse.

Step-by-Step Guidance

1. Label the points as $A = (−5, 3)$, $B = (6, 0)$, and $C = (5, 5)$.

2. Find the distances between each pair of points: $AB$, $BC$, and $AC$ using the distance formula.

3. Calculate the squares of each side length.

4. Check if the sum of the squares of the two shorter sides equals the square of the longest side.

Try solving on your own before revealing the answer!

Final Answer: Yes, the points form a right triangle.

After calculating all three side lengths and checking the Pythagorean Theorem, the condition is satisfied.