In △abc, m∠abc=40°, bl (l∈ ac ) is the angle bisector of ∠b. point m∈ ab so that lm ⊥ ab and n∈ bc so that ln ⊥ bc . find the measures of the angles of △abc if m∠cln=3m∠alm and prove that cn= 1/2 cl.

In △ABC, m∠ABC=40°, BL (L∈ AC ) is the angle bisector of ∠B.

Point M∈ AB so that LM ⊥ AB and N∈ BC so that LN ⊥ BC .

m∠CLN=3m∠ALM

Let m∠ALM = x so m∠CLN = 3x

According to the diagram,

m∠A = 90 - x and m∠c= 90 - 3x

As we know that sum of all the angles of a triangle is 180,

m∠A + m∠B + m∠C = 180

(90 - x) + 40 + (90 - 3x) = 180

220 - 4x = 180

-4x = -40

x = 10

So m∠A = 90 - x = 90 - 10 =80

m∠C = 90 - 3x = 90 - 30 = 60

In ΔCLN, ∠N= 90°, ∠C = 60° and ∠CLN = 30°

So by 30 -60-90 theorem

Side opposite to 90° = 1/2( side opposite to 30°)

CN = 1/2 (CL)

the value of v.w= -12.

step-by-step explanation:

we are given three vector r,v and w as:

r = < 5, -5, -2> ; v = < 2, -8, -8> ; w = < -2, 6, -5>

we know that r,v and w could also be represented as:

r=5i-5j-2k

v=2i-8j-8k

and w= -2i+6j-5k

where i,j and k are the unit vectors of x,y and z-axis respectively.

also we know that it has the following property:

i.i=1 ; j.j=1; k.k=1

and i.=j.i=i.k=k.i=j.k=k.j=0

i.e. when we multiply two vectors or we find it's dot product we need to only multiply the components of x coordinates with each other; y with each other and z with each other.

hence, v.w= (2i-8j- -2i+6j-5k)

                  = 2×(-2)+(-8)×(6)+(-8)×(-5)

                  = -4-48+40

                  = -12

hence, v.w= -12

total amount of money that he should pay if he has not used his duty free exemption is $98

step-by-step explanation:

given a united states citizen returning to the states declares the following items at the customs office: 3 shirts at $8.50 each 2 dresses at $27.50 each 1 pair of gold cuff links at $17.50 per pair. we have to find how much he pay

cost of 3 shirts if one cost at $8.50 is

cost of 2 dresses if one cost at $27.50 is  

cost of 1 pair of gold cuff at $17.50

so, total amount of money that he should pay if he has not used his duty free exemption is   $25.5+$55+$17.50=$98