In trapezoid ABCD, base AD = 4, BC = 2. Point K belongs to line AD, line CK divides the trapezoid into parts

| May 22, 2021 | education

In trapezoid ABCD, base AD = 4, BC = 2. Point K belongs to line AD, line CK divides the trapezoid into parts, the areas of which are 1: 3 (vertex B belongs to the smaller part). CK meets AB at point M. Find the length of the line segment parallel to the base of the trapezoid passing through point M.

triangle KCD = SABCD / 4 SABCD = (AD + BC) * h / 2 AD = 2BC SABCD = 3BC * h / 2 triangle KCD = 3BC * h / 8 triangle KCD = KD * h / 2 3BC * h / 8 = KD * h / 2 <=> KD = 3BC / 4 KD = 1.5 AK = 4-1.5 = 2.5 CM – median △ ACD: AM = 1 / 2AD = BC CM = AB Median on three sides: Mc = √ (2a ^ 2 + 2b ^ 2 – c ^ 2) / 2 CM = √ (2AC ^ 2 + 2CD ^ 2 – AD ^ 2) / 2 √7 = √ (2AC ^ 2 + 2CD ^ 2 – 4 ^ 2) / 2 <=> 7 = (AC ^ 2 + CD ^ 2) / 2 – 4 <=> AC ^ 2 + CD ^ 2 = 22 AD ^ 2 = AC ^ 2 + CD ^ 2 -2AC * CD * cos (ACD) 16 = AC ^ 2 + CD ^ 2 – AC * CD 16 = 22 – AC * CD <=> AC * CD = 6 S triangle ACD = AC * CD * sin (ACD) / 2 S triangle ACD = 3√3 / 2 S triangle ACD = AD * h / 2 3√3 / 2 = 4 * h / 2 <=> h = 3√3 / 4 S triangle ACK = AK * h / 2 S triangle ACK = 15√ 3/16



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