8. In right triangle LMN shown below, altitude MK is drawn to LN from M. IF LM = 12 and LK =10, then
which of the following is the length of KN?
(1) 2.8
(2) 4.4
(3) 6.2
(4) 14.4

The length of the KN is 4.4

Step-by-step explanation:

We know from Pythagoras theorem  

In a right angle ΔLMN  

Base² + perpendicular² = hypotenuse ²

From the properties of triangle we also know that altitudes are ⊥ on the sides they fall.

Hence ∠LKM = ∠NKM = 90 °

Given values-

LM=12

LK=10

Let KN be “s”

⇒LN= LK + KN

⇒LN= 10+x        eq 1

Coming to the Δ LKM

⇒LK²+MK²= LM²

⇒MK²= 12²-10²

⇒MK²= 44           eq 2

Now in Δ MKN

⇒MK²+ KN²= MN²

⇒44+s²= MN²       eq 3

In Δ LMN

⇒LM²+MN²= LN²

Using the values of MN² and LN² from the previous equations

⇒12² + 44+s²= (10+s) ²

⇒144+44+s²= 100+s²+20s

⇒188+s²= 100+s²+20s    cancelling the common term “s²”

⇒20s= 188-100

∴ s= 4.4

Hence the value of KN is 4.4  

7.9m and 4.3°

step-by-step explanation:

let east be +x and north be +y, x = distance * cos θ, y = distance * sin θ  

for 3.5 south, y = -3.5  

for 8.2, x = 8.2 * cos 30 and y = 8.2 * sin 30  

for 15 west, x = -15  

total x = 8.2 * cos 30 – 15 ≈ -7.9  

total y = -3.5 + 8.2 * sin 30 = 0.6  

magnitude = [(8.2 * cos 30 – 15)^2 + 0.6^2)]^0.5  

the magnitude is approximately 7.9 m  

tan θ = y/x = 0.6/-7.9  

the angle is approximately 4.3˚ north of west