Question

Triangle ABC

has vertices A(7, 2)
, B(−1, 8)
, and C(13, 10)
. Points P
and Q
are the mid-points of sides BC
and AC
respectively.



Enter a number in each box to complete the statement.





The length of the segment PQ
is
units.



The length of the segment AB
is
units.

Answer 1

The length of segment PQ is 5 units, and the length of segment AB is 10 units, calculated using the distance formula and the coordinates given for the vertices and midpoints of the triangle.

Step-by-step explanation:

To find the length of segment PQ, we first need to find the midpoints of sides BC and AC of triangle ABC. The midpoint of a segment with endpoints (x1, y1) and (x2, y2) is given by the coordinate pair ((x1 + x2)/2, (y1 + y2)/2).

Midpoint of BC (point P) = ((-1 + 13)/2, (8 + 10)/2)

= (6, 9)

Midpoint of AC (point Q) = ((7 + 13)/2, (2 + 10)/2)

= (10, 6)

Now, we use the distance formula to find the length of segment PQ: sqrt((x2 - x1)^2 + (y2 - y1)^2)

Length of PQ = sqrt((10 - 6)^2 + (6 - 9)^2)

= sqrt(16 + 9)

= sqrt(25)

= 5 units

To find the length of segment AB, we again use the distance formula:

Length of AB = sqrt((-1 - 7)^2 + (8 - 2)^2)

= sqrt(64 + 36)

= sqrt(100)

= 10 units.

Answer 2

The length of the segment PQ is 7.07 units.

The length of the segment AB is 10 units.

Finding the midpoints P and Q:

The midpoint of a segment is the point that lies exactly halfway between its two endpoints.

To find the coordinates of P, the midpoint of BC, we average the x-coordinates of B and C, and average their y-coordinates:

P = ((-1 + 13)/2, (8 + 10)/2) = (6, 9)

Similarly, to find the coordinates of Q, the midpoint of AC, we average the coordinates of A and C:

Q = ((7 + 13)/2, (2 + 10)/2) = (10, 6)

Calculating the lengths of PQ and AB using the distance formula:

The distance formula measures the distance between two points in a coordinate plane:

Distance = √((x2 - x1)² + (y2 - y1)²)

Applying the distance formula to find PQ:

PQ = √((10 - 6)² + (6 - 9)²) = √(16 + 9) = √25 = 7.07 units

Applying the distance formula to find AB:

AB = √((-1 - 7)² + (8 - 2)²) = √(64 + 36) = √100 = 10 units

Therefore, the length of segment PQ is 7.07 units, and the length of segment AB is 10 units.