Question

1. what is the measure of \\(\overline{qs}\\)?

• q to r: 6.2 mm
• r to s: 15.7 mm

options: 6.95 mm, 9.2 mm, 21.9 mm, 22.4 mm

1. use the number line.

find \\(my = ____.

1. find \\(x\\) if:

• \\(b\\) is between \\(a\\) and \\(c\\).
• \\(ab = x + 5\\);
• \\(bc = 2(x - 3)\\); and
• \\(ab = bc\\).

1. what is the distance between \\(a(-3, -4)\\) and \\(b(5, 1)\\)?

options: \\(\sqrt{256}\\), \\(\sqrt{130}\\), \\(\sqrt{32}\\)

Explanation:

Problem 5:

Step1: Identify the segments

QS is composed of QR and RS. QR is 6.2 mm and RS is 15.7 mm.

Step2: Add the lengths

To find QS, we add QR and RS: \( QS = QR + RS = 6.2 + 15.7 \)

Step3: Calculate the sum

\( 6.2 + 15.7 = 21.9 \)

Step1: Recall segment addition postulate

Since B is between A and C, \( AB + BC = AC \). Given \( AB = x + 5 \), \( BC = 2(x - 3) \), and \( AC = AB + BC \) (wait, actually, from the diagram, AC should be AB + BC, but also, maybe we can assume AC is the sum. Wait, the problem says "Find x if: B is between A and C; AB = x + 5; BC = 2(x - 3); AB = BC". Wait, maybe a typo, but if AB = BC, then \( x + 5 = 2(x - 3) \)

Step2: Solve the equation

\( x + 5 = 2x - 6 \)
Subtract x from both sides: \( 5 = x - 6 \)
Add 6 to both sides: \( x = 5 + 6 = 11 \)
Wait, but also, if we consider AC = AB + BC, but maybe the problem is AB = BC. Let's check again. The problem says "AB = x + 5; BC = 2(x - 3); AB = BC". So set them equal:
\( x + 5 = 2(x - 3) \)
\( x + 5 = 2x - 6 \)
\( 5 + 6 = 2x - x \)
\( x = 11 \)

Step1: Identify coordinates

Point A is (-3, -4) and point B is (4, 5).

Step2: Use distance formula

The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

Step3: Substitute coordinates

\( x_1 = -3, y_1 = -4, x_2 = 4, y_2 = 5 \)
\( d = \sqrt{(4 - (-3))^2 + (5 - (-4))^2} = \sqrt{(7)^2 + (9)^2} = \sqrt{49 + 81} = \sqrt{130} \)

Answer:

21.9 mm

Problem 7: