In △abc, cm is the median to ab and side bc is 12 cm long. there is a point p∈ cm and a line ap intersecting bc at point q. find the lengths of segments cq and bq , if p is the midpoint of cm

4 cm

Step-by-step explanation:

Draw the line MD that is parallel to AQ. Consider triangles BMD and BAQ. These triangles are similar, because they have three pairs of congruent angles (angles BMD and BAQ are corresponding, angles BDM and BQA are corresponding and angle B is common). The coefficient of similarity is because point M is midpoint of AB. Then

Since then

Consider triangles CPQ and CMD. These triangles are similar, because they have three pairs of congruent angles (angles CPQ and CMD are corresponding, angles CQP and CDM are corresponding and angle C is common). The coefficient of similarity is because point P is midpoint of CM. Then

Since then

Hence you get

step-by-step explanation:

30% of $420 is 0.30($420) = $126 and $420 + $126 = $546.

or, 130% of $420 is 1.3($420) = $546.

notice that this is not the $600 original list price with which we started.

why doesn't this work?

the method does not work because 30% of $420 is not the same amount as the 30% of $600. it may seem obvious when stated this way, but it is a very common error made on problems of this type. we cannot just use the same percent and get back to the original amount because we are taking that percent of a different value.

to work problems of this type correctly, it is generally best to write an equation for the situation and then solve.

example: since (100% – 30%)(original price) = (sale price)

70% of the   (original price) is $420.

0.7x   = 420

equation image indicator

x = 600

the original cost was $600.

we may also solve this problem using the two-step method. working the problem in two steps requires us to use more algebra skills. we have to know that x = 1x (identity property of multiplicationtext annotation indicator) and that 1x – 0.3x = 0.7x (subtracting like terms). then we solve the problem as follows:

(original price) – 30%(original price) = sale price

x – 0.3x = 420

0.7x = $420

equation image indicator

x = 600

the original cost was $600.  

note that with both of the above methods we do obtain the correct original amount.

example: after a 5% pay raise, hermione is earning $22,680 per year. what was she earning before the pay raise?

105% of (original salary) = current salary

1.05x = 22,680

equation image indicator

x = 21,600

hermione earned $21,600 per year before the pay raise.

this is an example so maybe you will get how to at least do it!

answer: 88

step-by-step explanation: