In △abc, point d∈ac with ad: dc=4: 3, point e∈bc so that be: ec=1: 5. if acde=5 in2, find abdc, aabd, and aabc.

A diagram can be helpful.

When triangles have the same altitude, their areas are proportional to their base lengths.

The altitude from D to line BC is the same for triangles BDC and EDC. The base lengths of these triangles have the ratio ...

... BC : EC = (1+5) : 5 = 6 : 5

so ABDC will be 6/5 times AEDC.

... ABDC = (6/5)×(5 in²)

... ABDC = 6 in²

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The altitude from B to line AC is the same for triangles BDC and BDA, so their areas are proportional to their base lengths. That is ...

... AABD : ABDC = AD : DC = 4 : 3

so AABD will be 4/3 times ABDC.

... AABD = (4/3)×(6 in²)

... AABD = 8 in²

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Of course, AABC is the sum of the areas of the triangles that make it up:

... AABC = AABD + ABDC = 8 in² + 6 in²

... AABC = 14 in²

for the first question it would be the second answer for the second question it would be the third answer and for the third question it would be the first answer

step-by-step explanation:

tell you what. i'll find the length of the medians and you can fill in the blanks where they belong.

br

b = (2a,0)

r = (a/2,b/2) remember these things are medians. they go to the 1/2 way point of the line opposite the vertex they face.

br^2 =   (2a - a/2)^2 + (0 - b/2)^2

br^2 = (3/2 a) ^2 + b^2 / 4

br^2 = 9/4 a^2 + b^2 / 4   we need to find some relationship between a and b.

let's try ab = bc  

ab = 2a

bc = (2a - a)^2 + (b - 0)^2

bc = sqrt(a^2 + b^2)

ab = bc

2a = sqrt(a^2 + b^2) square both sides.

4a^2 = a^2 + b^2 subtract a^2 from both sides.

sqrt(3a^2) = sqrt(b^2)

sqrt(3)a = b

let's put b into br^2

br^2 = 9/4 a^2 + 3b^2 / 4

br^2 = 12 a^2 / 4

br^2 = 3a^2 

br = sqrt(3) * a

cp

c = (a,b) ; p = (a,0) this is another application of the distance formula, and it is a good one.

cp^2 = (a - a)^2 + (b - 0)^2

cp^2 = 0 + b^2

cp^2 = b^2 which you can see from the diagram. 

cp = sqrt(b^2)

cp = b but b = sqrt(3) * a 

cp = sqrt(3)*a

aq

a = (0,0)

q = (3/2) a, b/2

aq^2 = (3a/2 -0)^2 + (b/2 - 0)^2

aq^2 = 9a^2/4 + b^2/4

but b = sqrt(3) * a 

aq^2 = 9a^2 / 4 + (sqrt(3)*a)^2 /4

aq^2 = 9 a^2/ 4 + 3 a^2 / 4

aq^2 = 12 a^2/4

aq^2 = 3 a^2

aq = (sqrt  3) * a which agrees with the other two.