SOLUTION: In the diagram below, AC is the median of triangle ABD. Find the area of triangle ABD Diagram: https://imgur.com/a/vr9cZmy

Algebra ->  Surface-area -> SOLUTION: In the diagram below, AC is the median of triangle ABD. Find the area of triangle ABD Diagram: https://imgur.com/a/vr9cZmy      Log On


   

Question 1149604: In the diagram below, AC is the median of triangle ABD. Find the area of triangle ABD
Diagram: https://imgur.com/a/vr9cZmy
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Given image

Let point E be the midpoint of segment BC. The red triangle AEC has unknown height h and unknown base x

Focusing solely on triangle AEC, we can use the pythagorean theorem to say

a%5E2+%2B+b%5E2+=+c%5E2

h%5E2+%2B+x%5E2+=+17%5E2

h%5E2+%2B+x%5E2+=+289

h%5E2+=+289-x%5E2 subtract x^2 from both sides

We'll use the last equation later on

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Mark triangle AED with another color. I'll use green. This triangle also has height h, but the base is now 3x

The 3x is from the fact that

EC = x
CD = BC = 2x
ED = EC+CD = x+2x = 3x

Once again, use the pythagorean theorem, but focus solely on triangle AED.

a%5E2+%2B+b%5E2+=+c%5E2

h%5E2+%2B+%283x%29%5E2+=+27%5E2

h%5E2+%2B+9x%5E2+=+729

289-x%5E2+%2B+9x%5E2+=+729 Replace h%5E2 with 289-x%5E2 (we're using h%5E2+=+289-x%5E2 found earlier)

289%2B8x%5E2+=+729

8x%5E2+=+729-289 Subtract 289 from both sides

8x%5E2+=+440

x%5E2+=+440%2F8 Divide both sides by 8

x%5E2+=+55

x+=+sqrt%2855%29 Apply square root to both sides

4x+=+4sqrt%2855%29 Multiply both sides by 4

The base of triangle ABD is exactly 4%2Asqrt%2855%29 units long

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Use that info to find the following value for h

h%5E2+=+289-x%5E2

h%5E2+=+289-55

h%5E2+=+234

h+=+sqrt%28234%29

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Now compute the area of triangle ABD

Area+=+%281%2F2%29%2A%28base%29%2A%28height%29

Area+=+%281%2F2%29%2A4sqrt%2855%29%2Asqrt%28234%29

Area+=+2%2Asqrt%2855%29%2Asqrt%28234%29

Area+=+2%2Asqrt%2855%2A234%29

Area+=+2%2Asqrt%2812870%29

Area+=+2%2Asqrt%289%2A1430%29

Area+=+2%2Asqrt%289%29%2Asqrt%281430%29

Area+=+2%2A3%2Asqrt%281430%29

Area+=+6%2Asqrt%281430%29 Exact area

Area+=+226.892 Approximate area

Units are in square cm, often written as cm%5E2