Question

find the length of each segment.
number line with points w, x, y; -5, 0, 5 marked; segments (overline{wx}) (question 5) and (overline{wy}) (question 6)

Explanation:

Problem 5: Length of \(\overline{WX}\)

Step 1: Determine positions of W and X

From the number line, let's assume each tick mark is 1 unit. Let's find the coordinates:

• Let's see, between -5 and 0, W is at -4? Wait, no, let's count the ticks. From -5, the next tick is W, then X, then to 0. Wait, the distance between -5 and 0 is 5 units, with how many ticks? Wait, from -5 to 0: the ticks are at -5, W, X, then 0? Wait, no, the number line: -5, then a tick (W), then a tick (X), then a tick, then 0? Wait, maybe each segment between ticks is 1 unit. Let's check Y is at 5. So from 0 to 5, there are 5 ticks? Wait, no, the number line has: -5, then W, then X, then three ticks to 0, then five ticks to 5? Wait, maybe better to find coordinates. Let's assume:

Looking at the number line:

• The position of W: Let's see, from -5, the next mark is W. Then X, then three marks to 0. So from -5 to 0: total 5 units (since -5 to 0 is 5). The number of segments: from -5 to W: 1 segment, W to X: 1 segment, X to 0: 3 segments. Wait, 1+1+3=5, so each segment is 1 unit. So W is at -5 + 1 = -4? Wait, no, -5, then W is at -4? Then X is at -4 + 1 = -3? Wait, no, maybe:

Wait, the distance from -5 to 0 is 5 units. The ticks: -5, W, X, (then three ticks to 0). Wait, maybe the coordinates:

Let's count the units between the points. Let's see, Y is at 5. So from 0 to 5, there are 5 units (since 5 - 0 = 5). So each tick mark is 1 unit. So:

• Position of W: Let's see, between -5 and 0. Let's count the ticks: -5, then W (1 unit right from -5: -5 + 1 = -4), then X (1 unit right from W: -4 + 1 = -3), then three units to 0 (so -3 + 3 = 0, which matches). Then from 0 to Y (5), there are 5 units (0 + 5 = 5). So:

Position of W: -4? Wait, no, maybe:

Wait, the number line: -5, then a tick (W), then a tick (X), then three ticks to 0. So from -5 to W: 1 unit, W to X: 1 unit, X to 0: 3 units. So total from -5 to 0: 1 + 1 + 3 = 5 units, which is correct (0 - (-5) = 5). So:

• Coordinate of W: -5 + 1 = -4? Wait, no, -5 to W: 1 unit, so W is at -5 + 1 = -4? Then X is at -4 + 1 = -3? Then from X to 0: 3 units, so -3 + 3 = 0, correct. Then from 0 to Y: 5 units, so Y is at 5.

Now, length of \(\overline{WX}\) is the absolute difference between coordinates of W and X.

Step 2: Calculate length of \(\overline{WX}\)

Coordinates:

• W: Let's confirm. Wait, maybe the ticks are: -5, then W (at -4), X (at -3), then -2, -1, 0, then 1, 2, 3, 4, 5 (Y at 5). Wait, no, from 0 to 5, there are 5 ticks (1,2,3,4,5), so Y is at 5. Then from -5 to 0: ticks at -5, -4 (W), -3 (X), -2, -1, 0. Wait, that's 5 units (from -5 to 0: 5 units, with ticks at -5, -4, -3, -2, -1, 0: 6 ticks, 5 segments). So each segment is 1 unit. So:

• W is at -4, X is at -3. Then length of \(\overline{WX}\) is \(|-3 - (-4)| = |1| = 1\)? Wait, no, that can't be. Wait, maybe I messed up the coordinates.

Wait, let's look again. The number line:

• The leftmost tick is -5, then W, then X, then three ticks to 0, then five ticks to Y at 5. Wait, maybe the distance between -5 and X is 2 units? Wait, maybe the coordinates are:

Let’s count the units between W and X. Let's see, from W to X: how many ticks? The diagram shows W and X with one tick between -5 and X? Wait, the original diagram:

The number line has:

• Left arrow, then tick at -5, then W, then a tick, then X, then three ticks to 0, then five ticks to Y at 5. Wait, maybe each small segment is 1 unit. So:

• From -5 to W: 1 unit (so W is at -5 + 1 = -4)
• From W to X: 1 unit (so X is at -4 + 1 = -3)
• From X to 0: 3 units (so 0 is at -3 + 3 = 0)
• From 0 to Y: 5 units (so Y is at 0 + 5 = 5)

So length of…

Step 1: Determine positions of W and Y

From the number line, Y is at 5, and W is at -4 (from previous step).

Step 2: Calculate length of \(\overline{WY}\)

Length is the absolute difference between coordinates of W and Y: \(|5 - (-4)| = |9| = 9\)? Wait, no, that can't be. Wait, maybe W is at -4, Y is at 5, so 5 - (-4) = 9? But let's check the number of segments. From W (-4) to 0: 4 units (since -4 to 0 is 4), then from 0 to Y (5): 5 units, so total 4 + 5 = 9. Yes, that makes sense.

Wait, but let's re-examine the number line. Let's count the ticks:

• -5, W, tick, X, tick, tick, 0, tick, tick, tick, tick, tick, Y (5). Wait, from -5 to 0: how many ticks? -5, W, tick, X, tick, tick, 0: that's 7 ticks, 6 segments? No, that can't be. Wait, maybe the number line is divided into 1-unit intervals, so each tick is 1 unit. So:

• -5, -4 (W), -3 (tick), -2 (X), -1 (tick), 0 (tick), 1 (tick), 2 (tick), 3 (tick), 4 (tick), 5 (Y). Wait, that would mean X is at -2, W is at -4. Then WX is |-2 - (-4)| = 2 units. Then WY is |5 - (-4)| = 9 units. Wait, that's different.

Wait, the user's diagram: "W" and "X" are dots, with a tick between -5 and X, and Y at 5. Let's look at the original problem again.

The number line:

• Left arrow, then tick at -5, then W (dot), then a tick, then X (dot), then three ticks to 0, then five ticks to Y (dot at 5). So:

• From -5 to W: 1 segment (1 unit)
• From W to X: 1 segment (1 unit)
• From X to 0: 3 segments (3 units)
• From 0 to Y: 5 segments (5 units)

So:

• W: -5 + 1 = -4
• X: -4 + 1 = -3
• 0: -3 + 3 = 0
• Y: 0 + 5 = 5

Therefore:

Problem 5: Length of \(\overline{WX}\)

Step 1: Find coordinates of W and X

• W is at -4 (1 unit right of -5)
• X is at -3 (1 unit right of W)

Step 2: Calculate distance

Distance = \(|-3 - (-4)| = |1| = 1\)

Answer:

1

Problem 6: Length of \(\overline{WY}\)