Question

Multiple Choice Which of the following represents the distance between the points with coordinates \((a, 0)\) and \((0, b) ? (A) \)\sqrt{2 a+2 b}\( (B) \)\sqrt{a^{2}+b^{2}}\( (c) \)\sqrt{a+b}\( (D) \)\sqrt{a^{2}-b^{2}}$

Short answer

The distance is given by option (B) \(\sqrt{a^2 + b^2}\).

Step-by-step solution

Identify Points Coordinates

The given points are \((a, 0)\) and \((0, b)\). The first point is located on the x-axis at \(a\), and the second point is on the y-axis at \(b\).

Apply the Distance Formula

The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}. \] Plug in the coordinates of the points: \(x_1 = a\), \(y_1 = 0\), \(x_2 = 0\), \(y_2 = b\).

Substitute the Coordinates

Substitute the coordinates into the distance formula: \[ d = \sqrt{(0 - a)^2 + (b - 0)^2} = \sqrt{(-a)^2 + b^2}. \]

Simplify the Expression

Simplify the expression: \[ d = \sqrt{a^2 + b^2}. \] Since both \((-a)^2\) and \(b^2\) simplify directly, you don't need any additional steps for simplification.

Match with Options Given

Compare the calculated expression \(\sqrt{a^2 + b^2}\) with the options provided: (A) \(\sqrt{2a + 2b}\), (B) \(\sqrt{a^{2}+b^{2}}\), (C) \(\sqrt{a+b}\), (D) \(\sqrt{a^{2}-b^{2}}\). Option (B) \(\sqrt{a^{2}+b^{2}}\) is the match.

Key concepts

Coordinates

Coordinates help us pinpoint the exact location of a point on a plane. In the Cartesian coordinate system, we use two numbers - typically described as an ordered pair (x, y) - to mark the position of a point. The first number in this pair denotes the position along the x-axis, which is horizontal, while the second number refers to the position along the y-axis, which is vertical.
When we talk about the coordinates of a point like (a, 0), it means the point is 'a' units away from the origin along the x-axis. Similarly, if we take (0, b), the point is 'b' units away from the origin along the y-axis. These numbers are really handy when you want to visually or numerically denote where something is, especially if you're dealing with geometry or algebra problems.

• Coordinates represent positions on a grid.
• The x-axis is horizontal; the y-axis is vertical.
• Coordinates are essential for plotting and measurements.

Point Distance Calculation

Calculating the distance between two points in the coordinate plane is crucial in geometry and other mathematical fields. The distance formula lets us find out how far apart two points are. This formula is based on the Pythagorean theorem and is expressed as:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
To effectively use this formula, you should substitute the coordinates of the points into it. For example, given points (a, 0) and (0, b), we replace (x_1, y_1) with (a, 0), and (x_2, y_2) with (0, b). This gives us the calculation:
\[ d = \sqrt{(0 - a)^2 + (b - 0)^2} = \sqrt{a^2 + b^2} \]
You can see from this calculation that the distance is dependent on the differences in x-values and y-values squared, then summed and square-rooted. Understanding how to apply this formula is very useful for accurately measuring the space between any two points with given coordinates.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operators. They form the building blocks of algebra and are used in various calculations and problem-solving processes. When working with coordinates and distances, you'll often see expressions like \(a^2 + b^2\) or \(\sqrt{a^2 + b^2}\).
These expressions allow us to manipulate and solve problems by substituting values and performing calculations.
Simplifying expressions is a crucial skill, helping to solve more elaborate mathematical problems succinctly. In the distance formula for our specific problem, \((0-a)^2 + (b-0)^2\) simplifies into \(a^2 + b^2\) because \((-a)^2 = a^2\) and \(b^2\) remains \(b^2\) when substituted.

• Algebraic expressions include terms formed by constants and variables.
• Simplifying expressions is essential for ease of calculation.
• Using algebraic techniques helps reveal answers to complex geometry questions easily.