Question

3 if the perimeter of an isosceles trapezoid is 50cm and one of its legs has a length of 12cm, then the median of this trapezoid has a length of ______ cm.
4 given that the area of a trapezoid is 22cm² and the height of it is 2cm, then the length of the midsegment of it is ______ cm.
5 as shown in the figure, in trapezoid abcd, $overline{ab}paralleloverline{cd}$. median $overline{ef}$ intersects diagonals $overline{ac}$ and $overline{bd}$ at point m and n. if $ef = 18$cm, and $mn = 8$cm, then $ab=$______. a. 10cm b. 13cm c. 20cm d. 26cm

Explanation:

Step1: Recall isosceles - trapezoid properties

In an isosceles trapezoid, the two non - parallel sides (legs) are equal. Let the length of each leg be \(l = 12\mathrm{cm}\), and the perimeter \(P=50\mathrm{cm}\). Let the sum of the lengths of the two bases be \(b_1 + b_2\). The perimeter formula is \(P=b_1 + b_2+2l\).

Step2: Calculate the sum of the bases

We know that \(P = 50\mathrm{cm}\) and \(l = 12\mathrm{cm}\). Substitute into the perimeter formula: \(50=b_1 + b_2+2\times12\). Then \(b_1 + b_2=50 - 24=26\mathrm{cm}\).

Step3: Use the median formula

The length of the median \(m\) of a trapezoid is given by \(m=\frac{b_1 + b_2}{2}\). Since \(b_1 + b_2 = 26\mathrm{cm}\), then \(m = 13\mathrm{cm}\).

Step4: Recall trapezoid area formula for mid - segment

The area formula of a trapezoid is \(A=\text{mid - segment}\times h\). Given \(A = 22\mathrm{cm}^2\) and \(h = 2\mathrm{cm}\). Let the length of the mid - segment be \(s\). Then \(A=s\times h\).

Step5: Calculate the mid - segment length

We have \(22=s\times2\). Solving for \(s\), we get \(s = 11\mathrm{cm}\).

Step6: Use the property of the trapezoid median and diagonals

In trapezoid \(ABCD\) with \(AB\parallel CD\) and median \(EF\), we know that \(EF=\frac{AB + CD}{2}\) and \(MN=\frac{|AB - CD|}{2}\). Let \(AB=x\) and \(CD = y\). We have \(EF = 18\mathrm{cm}\) and \(MN = 8\mathrm{cm}\), so \(\frac{x + y}{2}=18\) (i.e., \(x + y=36\)) and \(\frac{|x - y|}{2}=8\) (i.e., \(|x - y| = 16\)).
Since \(x>y\) (assuming \(AB\) is the longer base), \(x - y=16\). Combining with \(x + y=36\), we add the two equations: \((x - y)+(x + y)=16 + 36\).

Step7: Solve the system of equations

\(2x=52\), then \(x = 26\mathrm{cm}\), so \(AB = 26\mathrm{cm}\).

Answer:

1. \(13\)
2. \(11\)
3. D. \(26\mathrm{cm}\)