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

Studdy Solution

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}

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}

STEP 4

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

STEP 6

Calculate the sum inside the square root.
d(P,Q)=58d(P, Q) = \sqrt{58}So, the distance between points and Q is 58\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 formulaM=(x1+x22,y1+y22)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=(+162,−17+(−14)2)M = \left(\frac{ +16}{2}, \frac{-17 + (-14)}{2}\right)

STEP 9

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

STEP 10

Calculate the division.
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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