Solved on Oct 31, 2023

Find the distance between points $P(9,-17)$ and $Q(16,-14)$ , and the coordinates of the midpoint $M$ of the segment $PQ$ . Simplify the distance.

STEP 1

Assumptions1. The coordinates of point are (9,-17) . The coordinates of point Q are (16,-14)
3. We are using the standard Euclidean distance formula and midpoint formula

STEP 2

First, we need to find the distance between the two points and Q. We can do this by using the distance formula $d(P, Q) = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}$

STEP 3

Now, plug in the given values for the coordinates of points and Q into the distance formula.
$d(P, Q) = \sqrt{(16 -9)^2 + (-14 - (-17))^2}$

STEP 4

implify the expressions inside the square root.
$d(P, Q) = \sqrt{(7)^2 + (3)^2}$

STEP 5

Calculate the square of the numbers and add them together.
$d(P, Q) = \sqrt{(49) + (9)}$

STEP 6

Calculate the sum inside the square root.
$d(P, Q) = \sqrt{58}$ So, the distance between points and Q is $\sqrt{58}$ .

STEP 7

Next, we need to find the coordinates of the midpoint M of the segment PQ. We can do this by using the midpoint formula $M = \left(\frac{x1 + x2}{2}, \frac{y1 + y2}{2}\right)$

STEP 8

Now, plug in the given values for the coordinates of points and Q into the midpoint formula.
$M = \left(\frac{ +16}{2}, \frac{-17 + (-14)}{2}\right)$

STEP 9

implify the expressions inside the brackets.
$M = \left(\frac{25}{2}, \frac{-31}{2}\right)$

STEP 10

Calculate the division.
$M = (12.5, -15.5)$ So, the coordinates of the midpoint M of the segment PQ are (12.5, -15.5).

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Find the distance between points $P(9,-17)$ and $Q(16,-14)$ , and the coordinates of the midpoint $M$ of the segment $PQ$ . Simplify the distance.

STEP 1

Assumptions1. The coordinates of point are (9,-17) . The coordinates of point Q are (16,-14)
3. We are using the standard Euclidean distance formula and midpoint formula

STEP 2

First, we need to find the distance between the two points and Q. We can do this by using the distance formula $d(P, Q) = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}$

STEP 3

Now, plug in the given values for the coordinates of points and Q into the distance formula.
$d(P, Q) = \sqrt{(16 -9)^2 + (-14 - (-17))^2}$

STEP 4

implify the expressions inside the square root.
$d(P, Q) = \sqrt{(7)^2 + (3)^2}$

STEP 5

Calculate the square of the numbers and add them together.
$d(P, Q) = \sqrt{(49) + (9)}$

STEP 6

Calculate the sum inside the square root.
$d(P, Q) = \sqrt{58}$ So, the distance between points and Q is $\sqrt{58}$ .

STEP 7

Next, we need to find the coordinates of the midpoint M of the segment PQ. We can do this by using the midpoint formula $M = \left(\frac{x1 + x2}{2}, \frac{y1 + y2}{2}\right)$

STEP 8

Now, plug in the given values for the coordinates of points and Q into the midpoint formula.
$M = \left(\frac{ +16}{2}, \frac{-17 + (-14)}{2}\right)$

STEP 9

implify the expressions inside the brackets.
$M = \left(\frac{25}{2}, \frac{-31}{2}\right)$

STEP 10

Calculate the division.
$M = (12.5, -15.5)$ So, the coordinates of the midpoint M of the segment PQ are (12.5, -15.5).

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Find the distance between points P(9,−17)P(9,-17)P(9,−17) and Q(16,−14)Q(16,-14)Q(16,−14), and the coordinates of the midpoint MMM of the segment PQPQPQ. Simplify the distance.

STEP 1

Assumptions1. The coordinates of point are (9,-17) . The coordinates of point Q are (16,-14)3. We are using the standard Euclidean distance formula and midpoint formula

STEP 2

First, we need to find the distance between the two points and Q. We can do this by using the distance formulad(P,Q)=(x2−x1)2+(y2−y1)2d(P, Q) = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}d(P,Q)=(x2−x1)2+(y2−y1)2​

STEP 3

Now, plug in the given values for the coordinates of points and Q into the distance formula.d(P,Q)=(16−9)2+(−14−(−17))2d(P, Q) = \sqrt{(16 -9)^2 + (-14 - (-17))^2}d(P,Q)=(16−9)2+(−14−(−17))2​

STEP 4

implify the expressions inside the square root.d(P,Q)=(7)2+(3)2d(P, Q) = \sqrt{(7)^2 + (3)^2}d(P,Q)=(7)2+(3)2​

STEP 5

Calculate the square of the numbers and add them together.d(P,Q)=(49)+(9)d(P, Q) = \sqrt{(49) + (9)}d(P,Q)=(49)+(9)​

STEP 6

Calculate the sum inside the square root.d(P,Q)=58d(P, Q) = \sqrt{58}d(P,Q)=58​So, the distance between points and Q is 58\sqrt{58}58​.

STEP 7

Next, we need to find the coordinates of the midpoint M of the segment PQ. We can do this by using the midpoint formulaM=(x1+x22,y1+y22)M = \left(\frac{x1 + x2}{2}, \frac{y1 + y2}{2}\right)M=(2x1+x2​,2y1+y2​)

STEP 8

Now, plug in the given values for the coordinates of points and Q into the midpoint formula.M=(+162,−17+(−14)2)M = \left(\frac{ +16}{2}, \frac{-17 + (-14)}{2}\right)M=(2+16​,2−17+(−14)​)

STEP 9

implify the expressions inside the brackets.M=(252,−312)M = \left(\frac{25}{2}, \frac{-31}{2}\right)M=(225​,2−31​)

STEP 10

Calculate the division.M=(12.5,−15.5)M = (12.5, -15.5)M=(12.5,−15.5)So, the coordinates of the midpoint M of the segment PQ are (12.5, -15.5).

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Find the distance between points $P(9,-17)$ and $Q(16,-14)$ , and the coordinates of the midpoint $M$ of the segment $PQ$ . Simplify the distance.

Assumptions1. The coordinates of point are (9,-17) . The coordinates of point Q are (16,-14)
3. We are using the standard Euclidean distance formula and midpoint formula

First, we need to find the distance between the two points and Q. We can do this by using the distance formula $d(P, Q) = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}$

Now, plug in the given values for the coordinates of points and Q into the distance formula.
$d(P, Q) = \sqrt{(16 -9)^2 + (-14 - (-17))^2}$

implify the expressions inside the square root.
$d(P, Q) = \sqrt{(7)^2 + (3)^2}$

Calculate the square of the numbers and add them together.
$d(P, Q) = \sqrt{(49) + (9)}$

Calculate the sum inside the square root.
$d(P, Q) = \sqrt{58}$ So, the distance between points and Q is $\sqrt{58}$ .

Next, we need to find the coordinates of the midpoint M of the segment PQ. We can do this by using the midpoint formula $M = \left(\frac{x1 + x2}{2}, \frac{y1 + y2}{2}\right)$

Now, plug in the given values for the coordinates of points and Q into the midpoint formula.
$M = \left(\frac{ +16}{2}, \frac{-17 + (-14)}{2}\right)$

implify the expressions inside the brackets.
$M = \left(\frac{25}{2}, \frac{-31}{2}\right)$

Calculate the division.
$M = (12.5, -15.5)$ So, the coordinates of the midpoint M of the segment PQ are (12.5, -15.5).