An isosceles trapezoid MNQP has parallel sides MN = 120 m and PQ = 200 m, and non-parallel sides MP = NQ = 50 m. Calculate the lengths of the segments on the base and the height of the trapezoid.

Step 1: Determine the lengths of the segments on the base. The given figure is an isosceles trapezoid MNQP with parallel sides MN and PQ. Given lengths: MN = 120 m PQ = 200 m MP = 50 m NQ = 50 m Draw perpendiculars from M and N to the base PQ, meeting PQ at points R and S respectively. This forms a rectangle MRSN and two right-angled triangles MPR and NQS. Since MRSN is a rectangle, RS = MN = 120 m. In an isosceles trapezoid, the segments PR and SQ are equal. The total length of the base PQ is the sum of these segments: PQ = PR + RS + SQ 200 m = PR + 120 m + SQ Let PR = SQ = x. 200 m = x + 120 m + x 200 m = 2x + 120 m Subtract 120 m from both sides: 2x = 200 m - 120 m 2x = 80 m Divide by 2: x = 80 m2 x = 40 m So, PR = 40 m and SQ = 40 m. Step 2: Calculate the height (h) of the trapezoid. Consider the right-angled triangle MPR. The hypotenuse is MP = 50 m. One leg is PR = 40 m. The other leg is MR, which is the height h. Using the Pythagorean theorem (a^2 + b^2 = c^2): MP^2 = MR^2 + PR^2 (50 m)^2 = h^2 + (40 m)^2 2500 m^2 = h^2 + 1600 m^2 Subtract 1600 m^2 from both sides: h^2 = 2500 m^2 - 1600 m^2 h^2 = 900 m^2 Take the square root of both sides: h = sqrt(900 m)^2 h = 30 m The height of the trapezoid is 30 m. Step 3: Calculate the area of the trapezoid. The formula for the area of a trapezoid is A = (1)/(2)(a+b)h, where a and b are the lengths of the parallel sides and h is the height. Here, a = MN = 120 m, b = PQ = 200 m, and h = 30 m. A = (1)/(2)(MN + PQ)h A = (1)/(2)(120 m + 200 m)(30 m) A = (1)/(2)(320 m)(30 m) A = (160 m)(30 m) A = 4800 m^2 The area of the trapezoid is 4800 m^2. Step 4: Calculate the perimeter of the trapezoid. The perimeter is the sum of the lengths of all its sides. P = MN + NQ + QP + PM P = 120 m + 50 m + 200 m + 50 m P = 420 m The perimeter of the trapezoid is 420 m. What's next? Send 'em! 📸