Question

If A = (8, −10) and B = (−4, 6), find the length of AB. 1 If MN = `1/2` AB, where M = (k, 5) and N = (4, −3), find the value of k.

Answer 1

Given:

Points A = (8, −10) and B = (−4, 6)

Points M = (k, 5) and N = (4, −3)

MN = `1/2` AB

Find the length of AB 

\[AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

\[A = (8, −10), \quad B = (−4, 6)\]

\[AB = \sqrt{(-4 - 8)^2 + (6 - (-10))^2}\]

= \[\sqrt{(-12)^2 + (16)^2}\]

= \[\sqrt{144 + 256}\] 

= \[\sqrt{400}\]

= 20 

Find the length of AB. If MN = `1/2`​ AB

Coordinates:

M = (k, 5), N = (4, − 3)

Length:

\[MN = \sqrt{(4 - k)^2 + (-3 - 5)^2} \]

= \[\sqrt{(4 - k)^2 + (-8)^2}\]

= \[\sqrt{(4-k)^2 + 64}\]

Use given condition. MN = `1/2` AB

\[MN = \frac{1}{2} \times 20 = 10\]

Therefore,

\[\sqrt{(4-k)^2 + 64} = 10\]

If A = (8, –10) and B = (–4, 6), find the length of AB. If MN = `1/2`​ AB, where M = (k, 5) and N = (4, –3), find the value of k.

(4 − k)² + 64 = 100

(4 − k)² = 36

4 − k = pm 6

If 4 − k = 6, then k = 4 − 6 = −2

If  4 − k = −6, then k = 4 + 6 = 10