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Big Ideas Math Integrated I, 2016

BI

Big Ideas Math Integrated I, 2016 View details

4. Proofs with Perpendicular Lines

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Exercise 2 Page 524

The distance from a point to a line is the length of the perpendicular segment from the point to the line. Therefore, the first and fourth statements both refer to the length of $\overline{X Y} .$ The only statement that does not refer to the distance between the point and the line is the second one.

To find the distance between the points, we use the Distance Formula. We will start with the distance

X Y .

d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

SubstitutePoints

Substitute $(3, 1)$ & $(- 3, 3)$

d_{X Y} = (3 - (- 3))^{2} + (1 - 3)^{2}

▼

Simplify RHS

$a - (- b) = a + b$

d_{X Y} = (3 + 3)^{2} + (1 - 3)^{2}

Add and subtract terms

d_{X Y} = 6^{2} + (- 2)^{2}

NegBaseToPosBase

$(- a)^{2} = a^{2}$

d_{X Y} = 6^{2} + 2^{2}

Calculate power

d_{X Y} = 36 + 4

Add terms

d_{X Y} = 40

Use a calculator

d_{X Y} = 6.324555 \dots

Round to $1$ decimal place(s)

d_{X Y} = 6.3

Let's also find the distance between

X

and

Z .

d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

SubstitutePoints

Substitute $(4, 4)$ & $(- 3, 3)$

d_{X Z} = (4 - (- 3))^{2} + (4 - 3)^{2}

▼

Simplify RHS

$a - (- b) = a + b$

d_{X Z} = (4 + 3)^{2} + (4 - 3)^{2}

Add and subtract terms

d_{X Z} = 7^{2} + 1^{2}

Calculate power

d_{X Z} = 49 + 1

Add terms

d_{X Z} = 50

Use a calculator

d_{X Z} = 7.071067 \dots

Round to $1$ decimal place(s)

d_{X Z} \approx 7.1

Subchapter links

Communicate Your Answer p.519

Monitoring Progress p.520-523

Exercises p.524-526