just need help on these last two problems~~

[xcrush]

I'm having trouble with these two problems.. can anyone help?

1. In the isosceles triangle ABC, AB = AC = 6 cm and BC = 10 cm. Point P is placed on the segment BC and points M and N are on the lines AB and AC respectively, so that PM is perpendicular to AB and PN is perpendicular to AC. Compute PM + PN.

2. The Pentasect is an ancient cult that counts in base five and believes that numbers that contain a 2 in their base five representation are evil. All other numbers are good. In addition, a number "n" is balanced if the number of good and evil numbers between 1 and "n" (inclusive) are equal. The smallest balanced number is 2 (since 1 is good and 2 is evil). Compute the largest balanced number. Give your answer in base 10.

 

[tkhunny]

Some idea concerning your present attempts would be the best place to start What have you done to solve them?

 

[stapel]

1) I'm going to add some labelling and fix some orientation, so we're looking at the same picture, okay?

Draw the triangle, putting "A" at the peak, "B" to the bottom left, and "C" to the bottom right. Since ABC is isosceles, then the measures of the angles at B and at C are equal. Call these angles "ß" (that is, "beta").

Draw the points P, M, and N, and indicate the right angles.

Draw (lightly) the height line for ABC, indicating that this line is perpendicular to BC. Label the new angles, formed at A, as "theta". (I can't draw the "theta" character here, so I'll refer to this as "@".)

Label the length between M and B as "m", the length between N and C as "n", and the length between B and P as "p". Then |AM| = 6 - m, |AN| = 6 - n, and |CP| = 10 - p.

Note that, since BMP is a right triangle sharing angle ß, that it must be similar to the right angle forming half of the isosceles triangle. Note that you can conclude the same thing about CNP. Form the similar-triangle proportionality statements.

From these equivalencies, you should be able to form an equation, giving |PM| + |PN| in terms of the other sides values. The variables should drop out, giving you a numerical value.

2) Sorry; no clue. Do we even know there is a biggest such number? What sorts of number-theory formulas and rules do you have?

Eliz.

 

[xcrush]

I still can't get the first question.. I did the similiar triangle proportions but can't seem to get an equation that works.

Also for #2.. what exactly is base 5 and base 10?

 

[stapel]

I still can't get the first question.. I did the similiar triangle proportions but can't seem to get an equation that works.

Please reply showing what you have tried. Thank you.

Also for #2.. what exactly is base 5 and base 10?

We can't really teach classes here. Are you familiar with number bases at all, or are you needing lessons on the topic? Please clarify. Thank you.

Eliz.