A segment in the coordinate plane has one endpoint at A(1, 9) and its midpoint at M(5, 6).… — ACT

A segment in the coordinate plane has one endpoint at $A(1, 9)$ and its midpoint at $M(5, 6)$ . Let $B$ represent the other endpoint of the segment. What is the distance between point $B$ and the point $C(3, -5)$ ?

A
$14$
B
$2\sqrt{10}$
$10$ Cevap
D
$5\sqrt{5}$
E
$\sqrt{5}$

Cevap

The distance between point

B

and point

C

10

First, we find the coordinates of point

B(x, y)

using the midpoint formula. Since

M(5, 6)

is the midpoint between

A(1, 9)

and

B(x, y)

, we set up the equations:

5 = \frac{1+x}{2}

which simplifies to

10 = 1+x

and

x = 9

; and

6 = \frac{9+y}{2}

which simplifies to

12 = 9+y

and

y = 3

. Thus, point

B

(9, 3)

. Next, we apply the distance formula to find the distance between

B(9, 3)

and

C(3, -5)

d = \sqrt{(9-3)^2 + (3-(-5))^2} = \sqrt{6^2 + 8^2} = \sqrt{36+64} = \sqrt{100} = 10

. This is the correct distance.

Adım Adım Çözüm

Find the coordinates of endpoint

B(x_B, y_B)

using the midpoint formula.

x_B = 9

and

y_B = 3

, so point

B

(9, 3)

Since

M(5, 6)

is the midpoint of segment

AB

, we can solve for

B

's coordinates using

5 = \frac{1 + x_B}{2}

and

6 = \frac{9 + y_B}{2}

Calculate the distance between

B(9, 3)

and

C(3, -5)

using the distance formula.

d = 10

Applying the distance formula:

d = \sqrt{(9-3)^2 + (3-(-5))^2} = \sqrt{6^2 + 8^2} = \sqrt{100} = 10

Anahtar Kavram

Using the midpoint formula to find a missing endpoint, and then applying the distance formula between two coordinate points.

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

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