Answer to Question #87448 in Geometry for Sayem

Question

1.In triangle ABC, side AB has length 10, and the A- and B-medians have length 9 and 12, respectively.
Compute the area of the triangle.

2. Jay is given 99 stacks of blocks, such that the i-th stack has i^2 blocks. Jay must choose a positive
integer N such that from each stack, he may take either 0 blocks or exactly N blocks. Compute the
value Jay should choose for N in order to maximize the number of blocks he may take from the 99
stacks.

3.Let ABCD be a square with side length 4. Consider points P and Q on segments AB and BC,
respectively, with BP = 3 and BQ = 1. Let R be the intersection of AQ and DP. If BR^2 can be
expressed in the form
m/n for coprime positive integers m, n, compute m + n.
please ans fast its urgent...

Answer 1

$$1.\ \Delta ABC: AB=10, AM=9, BK=12, AK=KC, BM=CM$$

$$AD : DM=2:1, AD=6, DM=3$$ $$BD : DK=2:1, BD=8, DK=4$$

$$\Delta ABD: p={10+6+8 \over 2}=12$$ $$\Delta ABD: Area-1=\sqrt{12(12-10)(12-6)(12-8)}=24$$

$$\Delta ABK: Area_2={3 \over 2}*Area_1={3 \over 2}*24=36$$

$$\Delta ABC: Area_3=2*Area_2=2*36=72$$

$$2.\ \displaystyle\sum_{i=1}^ki^2={k(k+1)(2k+1) \over 6}$$

$$N=\displaystyle\sum_{i=1}^{99}i^2={99(99+1)(2*99+1) \over 6}=328350$$

$$3.\ A(0, 0), B(0, 4), C(4, 4), D(4, 0), P(0, 1), Q(1, 4).$$

$$AQ: y=kx; k={4-0 \over 1-0}=4; y=4x$$

$$DP: y=mx+b, m={0-1 \over 4-0}={-1 \over 4}, b=1, y=-{1 \over 4}x+1$$

$$R: 4x=-{1 \over 4}x+1 \Rightarrow x={4 \over 17}, y={16 \over 17}$$

$${BR}^2=(0-{4 \over 17})^2+(4-{16\over 17})^2={2720 \over289}={160 \over17}={m \over n}$$

$$m+n=160+17=177$$