OlympiadHardGeometry7–9

Triangle $ABC$ has area $72$ cm². The point $D$ lies on $BC$ with $BD : DC = 1 : 2$ , and the point $E$ lies on $AC$ with $AE : EC = 1 : 3$ . What is the area of triangle $ADE$ , in cm²?

O triângulo $ABC$ tem área $72$ cm². O ponto $D$ está em $BC$ com $BD : DC = 1 : 2$ , e o ponto $E$ está em $AC$ com $AE : EC = 1 : 3$ . Qual é a área do triângulo $ADE$ , em cm²?

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Solution

Step 1 of 3

1.Triangles $ADC$ and $ABC$ share the height from $A$ , so their areas are in the ratio $DC : BC = 2 : 3$ , giving $[ADC] = 48$ cm².

Triangle ABCABCABC has area 727272 cm². The point DDD lies on BCBCBC with BD:DC=1:2BD : DC = 1 : 2BD:DC=1:2, and the point EEE lies on ACACAC with AE:EC=1:3AE : EC = 1 : 3AE:EC=1:3. What is the area of triangle ADEADEADE, in cm²?

Answer options

Triangle $ABC$ has area $72$ cm². The point $D$ lies on $BC$ with $BD : DC = 1 : 2$ , and the point $E$ lies on $AC$ with $AE : EC = 1 : 3$ . What is the area of triangle $ADE$ , in cm²?