# A straight line is drawn through vertex C of the trapezoid ABCD, parallel to the lateral side AB. It crosses the large

A straight line is drawn through vertex C of the trapezoid ABCD, parallel to the lateral side AB. It crosses the large base AD at point K. The perimeter of the trapezoid ABCD is 37cm, DK = 6cm, AK = 9cm. Calculate: a) the length of the midline of the trapezoid; b) the perimeter of the triangle KCD.

Since the bases of the trapezoid BC and AD are parallel, then AK || BC.

AB || CK – by condition.

Consequently, the quadrangle АBCК is a parallelogram, since its opposite sides are parallel in pairs, which means that AK = BC = 9 cm and AB = CK.

a) The middle line of the trapezoid is equal to the half-sum of its bases:

l = (AD + BC) / 2 = (AK + KD + BC) / 2 = (9 + 6 +9) / 2 = 12 cm.

b) The perimeter of a trapezoid is equal to the sum of the lengths of its sides: PABCD = AB + BC + CD + AD.

Hence, AB + CD = PABCD – BC – AD = PABCD – BC – AK – KD = 37 – 9 – 9 – 6 = 13 cm.

Since AB = CK, then AB + CD = CK + CD = 13 cm.

Find the perimeter of the triangle KCD: PKCD = CK + CD + KD = 13 + 6 = 19 cm.

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