Question

8.03
problems 1-2: here is the line segment gh. each grid square represents 1 square unit. use the ruler, circle, or square if they help with your thinking.

1. determine the approximate length of gh.
2. determine the exact length of gh.

ruler
circle
square

1. determine the value of $sqrt{16}$.

problems 4-5: estimate each square root. draw a square if it helps with your thinking.

1. $sqrt{35}$
2. $sqrt{66}$

problems 6-7: determine which two whole numbers each square root is between.

1. $sqrt{7}$
2. $sqrt{31}$

Explanation:

Step1: Identify coordinates of G, H

Point $G=(1,5)$, Point $H=(5,1)$

Step2: Calculate horizontal/vertical distances

Horizontal difference: $5-1=4$; Vertical difference: $5-1=4$

Step3: Approximate length via ruler

Visually, the ruler aligns to ~5.6 units

Step4: Exact length via distance formula

$$\text{Length} = \sqrt{(5-1)^2 + (1-5)^2} = \sqrt{16+16} = \sqrt{32} = 4\sqrt{2}$$

Step5: Evaluate $\sqrt{16}$

$\sqrt{16} = 4$ (since $4^2=16$)

Step6: Estimate $\sqrt{35}$

$5^2=25$, $6^2=36$, so $\sqrt{35}\approx5.9$

Step7: Estimate $\sqrt{66}$

$8^2=64$, $9^2=81$, so $\sqrt{66}\approx8.1$

Step8: Find bounds for $\sqrt{7}$

$2^2=4$, $3^2=9$, so between 2 and 3

Step9: Find bounds for $\sqrt{31}$

$5^2=25$, $6^2=36$, so between 5 and 6

Answer:

1. Approximate length of $GH$: $\approx 5.6$ units
2. Exact length of $GH$: $4\sqrt{2}$ units
3. $\sqrt{16} = 4$
4. $\sqrt{35} \approx 5.9$
5. $\sqrt{66} \approx 8.1$
6. $\sqrt{7}$ is between 2 and 3
7. $\sqrt{31}$ is between 5 and 6