Find the distance between the pair of points and . If necessary, express the answer in simplified radical form and then round
The distance between the given points is units.
(Simplify your answer. Type an exact answer, using radicals as needed.)
Express the distance rounded to two decimal places if the exact simplified answer contains radicals.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The distance between the given points is approximately units.
(Type an integer or decimal rounded to two decimal placessas needed.)
B. The exact simplified form does not contain radicals.

Ask by Lang Schofield. in the United States

Mar 22,2025

Question

Find the distance between the pair of points and . If necessary, express the answer in simplified radical form and then round
The distance between the given points is units.
(Simplify your answer. Type an exact answer, using radicals as needed.)
Express the distance rounded to two decimal places if the exact simplified answer contains radicals.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The distance between the given points is approximately units.
(Type an integer or decimal rounded to two decimal placessas needed.)
B. The exact simplified form does not contain radicals.

Ask by Lang Schofield. in the United States

Mar 22,2025

UpStudy AI · Accepted Answer

**Step 1: Use the Distance Formula**

The distance $$ d $$ between two points $$(x_1, y_1)$$ and $$(x_2, y_2)$$ is given by

$$
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.
$$

**Step 2: Substitute the Given Points**

Let $$(x_1, y_1) = (4, 5)$$ and $$(x_2, y_2) = (7, 8)$$. Substitute these into the formula:

$$
d = \sqrt{(7 - 4)^2 + (8 - 5)^2}.
$$

**Step 3: Simplify the Expression**

Calculate the differences:

$$
7 - 4 = 3 \quad \text{and} \quad 8 - 5 = 3.
$$

Square the differences:

$$
3^2 = 9 \quad \text{and} \quad 3^2 = 9.
$$

So,

$$
d = \sqrt{9 + 9} = \sqrt{18}.
$$

**Step 4: Express the Distance in Simplified Radical Form**

Simplify $$\sqrt{18}$$ by factoring:

$$
\sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9}\sqrt{2} = 3\sqrt{2}.
$$

Thus, the exact distance is

$$
d = 3\sqrt{2} \text{ units}.
$$

**Step 5: Round the Distance to Two Decimal Places**

Calculate the approximate value of $$3\sqrt{2}$$:

$$
3\sqrt{2} \approx 3 \times 1.414 \approx 4.242.
$$

Rounded to two decimal places, the distance is approximately

$$
4.24 \text{ units}.
$$

**Final Answer Choice:**

A. The distance between the given points is approximately $$4.24$$ units.
The distance between the points is approximately 4.24 units.