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What is the distance between the points (-2, 2) and (1, 6)?

Mathematics

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Answer

B

5 units

Step-by-step solution

11 steps

Part 11· 4 steps

To find the distance between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2), we use the distance formula:

D=(x2−x1)2+(y2−y1)2D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

  1. Identify the coordinates.

    Given points are (−2,2)(-2, 2) and (1,6)(1, 6). So, x1=−2x_1 = -2, y1=2y_1 = 2, x2=1x_2 = 1, y2=6y_2 = 6.

  2. Substitute the values into the distance formula.

    D=(1−(−2))2+(6−2)2D = \sqrt{(1 - (-2))^2 + (6 - 2)^2} D=(1+2)2+(4)2D = \sqrt{(1 + 2)^2 + (4)^2} D=(3)2+(4)2D = \sqrt{(3)^2 + (4)^2}

  3. Calculate the squares and sum them.

    D=9+16D = \sqrt{9 + 16} D=25D = \sqrt{25}

  4. Find the square root.

    D=5D = 5

    The distance is 5 units.

    The correct answer is b) 5 units.

Part 12

Angles that sum up to 90∘90^\circ are defined as complementary angles.

The correct answer is a) complementary angles.

Part 13

An infinite number of lines can pass through a single point. Imagine drawing lines through the origin (0,0)(0,0) at every possible angle; there are infinitely many such lines.

The correct answer is d) ∞\infty.

Part 14· 3 steps

To find the midpoint of two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2), we use the midpoint formula:

M=(x1+x22,y1+y22)M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)

  1. Identify the coordinates.

    Given points are (−2,8)(-2, 8) and (8,−2)(8, -2). So, x1=−2x_1 = -2, y1=8y_1 = 8, x2=8x_2 = 8, y2=−2y_2 = -2.

  2. Substitute the values into the midpoint formula.

    M=(−2+82,8+(−2)2)M = \left(\frac{-2 + 8}{2}, \frac{8 + (-2)}{2}\right)

  3. Calculate the sums and divide by 2.

    M=(62,62)M = \left(\frac{6}{2}, \frac{6}{2}\right) M=(3,3)M = (3, 3)

    The coordinates of the midpoint are (3,3)(3, 3).

    The correct answer is b) (3,3)(3,3).

Part 15· 4 steps

Given the equation x+1x=2x + \frac{1}{x} = 2. We need to find the value of x2+1x2x^2 + \frac{1}{x^2}.

  1. Square both sides of the given equation.

    (x+1x)2=22\left(x + \frac{1}{x}\right)^2 = 2^2

  2. Expand the left side using the formula (a+b)2=a2+2ab+b2(a+b)^2 = a^2 + 2ab + b^2.

    x2+2⋅x⋅1x+(1x)2=4x^2 + 2 \cdot x \cdot \frac{1}{x} + \left(\frac{1}{x}\right)^2 = 4

  3. Simplify the equation.

    x2+2+1x2=4x^2 + 2 + \frac{1}{x^2} = 4

  4. Isolate x2+1x2x^2 + \frac{1}{x^2}.

    x2+1x2=4−2x^2 + \frac{1}{x^2} = 4 - 2 x2+1x2=2x^2 + \frac{1}{x^2} = 2

    The value of x2+1x2x^2 + \frac{1}{x^2} is 2.

    The correct answer is d) 2.

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