In the standard (x, y) coordinate plane, the midpoint of the line segment with endpoints A… — ACT

In the standard $(x, y)$ coordinate plane, the midpoint of the line segment with endpoints $A(1, 3)$ and $B\left(7, \frac{x}{2}\right)$ is $M$ . If the distance from the origin $(0, 0)$ to $M$ is $\sqrt{\frac{89}{4}}$ units, and $x$ is a positive number, what is the value of $x$ ?

A
2
4Cevap
C
8
D
14
E
16

Cevap

The correct value of

x

4

The midpoint

M

A(1, 3)

and

B\left(7, \frac{x}{2}\right)

has coordinates

\left(\frac{1+7}{2}, \frac{3 + \frac{x}{2}}{2}\right) = \left(4, \frac{x+6}{4}\right)

. The distance from

(0, 0)

M

\sqrt{4^2 + \left(\frac{x+6}{4}\right)^2} = \sqrt{\frac{89}{4}}

. Squaring both sides yields

16 + \frac{(x+6)^2}{16} = \frac{89}{4}

. Multiplying through by 16 gives

256 + (x+6)^2 = 356

, which simplifies to

(x+6)^2 = 100

. Taking the square root of both sides gives

x+6 = 10

(since

x

is positive), which simplifies to

x = 4

Adım Adım Çözüm

Find the coordinates of the midpoint

M(x_M, y_M)

of the segment

AB

M = \left(4, \frac{x+6}{4}\right)

By the midpoint formula,

x_M = \frac{1 + 7}{2} = 4

and

y_M = \frac{3 + \frac{x}{2}}{2} = \frac{\frac{6+x}{2}}{2} = \frac{x+6}{4}

Set up the distance equation from the origin

(0, 0)

M\left(4, \frac{x+6}{4}\right)

using the distance formula.

\sqrt{4^2 + \left(\frac{x+6}{4}\right)^2} = \sqrt{\frac{89}{4}}

The distance between the origin

(0, 0)

and any point

(x, y)

is given by

d = \sqrt{x^2 + y^2}

Solve the distance equation for the positive variable

x

x = 4

Squaring both sides gives

16 + \frac{(x+6)^2}{16} = \frac{89}{4}

. Multiplying by 16 yields

256 + (x+6)^2 = 356

, which simplifies to

(x+6)^2 = 100

. Since

x

is positive,

x+6 = 10 \implies x = 4

Anahtar Kavram

Solving coordinate geometry problems by combining the midpoint formula and the distance formula.

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Konular

Bu konudan daha fazla soru