Kai K

21 Juli 2024 09:42

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Kai K

21 Juli 2024 09:42

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Let ABC be a triangle with AB = 18, BC = 24, and CA = 20. D is placed on AB such that AD = 15. E is placed on BC such that EC = 20. Call the intersection of lines AE and DC point F. Compute: (area of triangle DEF)^2 = (7^3 * 6^3)/(5^3)

Let ABC be a triangle with AB = 18, BC = 24, and CA = 20. D is placed on AB such that AD = 15. E is placed on BC such that EC = 20. Call the intersection of lines AE and DC point F. Compute: (area of triangle DEF)^2 = (7^3 * 6^3)/(5^3)

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<p><strong>Step 1: Calculate the area of triangle ABC</strong></p><p>Using Heron's formula, the semi-perimeter <i><sub>s</sub> =<sup>18+24+20 / 2 = 31. </sup></i><sup>The area ∆ of triangle ABC is:</sup></p><p><sup>∆ = </sup>√ s(s-a) (s-b) (s-c) = √31 (31-18) (31-24) (31-20) = √31.13.7.11 = √31031 = 165</p><p><strong>Step 2: Use mass points to find ration</strong></p><p>Assign mass points: <i>m</i>(A)= 24, <i>m</i>(B)= 20, <i>m</i>(C)= 18. since AD= 15 and DB= 3, <i>m</i>(D)= 20 + 24= 44. simiarly, <i>m</i>(E)= 18+20= 38.</p><p><strong>Step 3: Find the coordinates of point F</strong></p><p>Using the mass points, the ratio <i>AF/FE = m(E)/m(A) </i>= <i>38/24 = 19/12, and DF/FC = m(C)/m(D) = 18/44 = 9/22.</i></p><p><strong>Step 4: Calculate the area of triangle DEF</strong></p><p>Using the rations, the area of triangle DEF is:</p><p><i>ext Area of DEF = 19/19+12 • 9/9+22 • 165 = 19/31 • 19/31 • 165= 19 • 9 •165/31² = 28215/961 = 29.375</i></p><p><strong>Step 5: Compute the final expression</strong></p><p><i><strong>(</strong>ext Area of DEF)² • 7³ • 6³/ 5³= 29.375² • 7³ • 6³/ 5³ = 862.5 • 2058/125 = 862.5 • 16.464 = 14190</i></p><p><i><strong><u>FINAL ANSWER : 14190.</u></strong></i></p>

Step 1: Calculate the area of triangle ABC

Using Heron's formula, the semi-perimeter s =18+24+20 / 2 = 31. The area ∆ of triangle ABC is:

∆ = √ s(s-a) (s-b) (s-c) = √31 (31-18) (31-24) (31-20) = √31.13.7.11 = √31031 = 165

Step 2: Use mass points to find ration

Assign mass points: m(A)= 24, m(B)= 20, m(C)= 18. since AD= 15 and DB= 3, m(D)= 20 + 24= 44. simiarly, m(E)= 18+20= 38.

Step 3: Find the coordinates of point F

Using the mass points, the ratio AF/FE = m(E)/m(A) = 38/24 = 19/12, and DF/FC = m(C)/m(D) = 18/44 = 9/22.

Step 4: Calculate the area of triangle DEF

Using the rations, the area of triangle DEF is:

ext Area of DEF = 19/19+12 • 9/9+22 • 165 = 19/31 • 19/31 • 165= 19 • 9 •165/31² = 28215/961 = 29.375

Step 5: Compute the final expression

(ext Area of DEF)² • 7³ • 6³/ 5³= 29.375² • 7³ • 6³/ 5³ = 862.5 • 2058/125 = 862.5 • 16.464 = 14190

FINAL ANSWER : 14190.

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