In the MNKP trapezoid, Angle M = 45 degrees, Angle P = 30 degrees, the sides are 8 cm and 10 cm

| August 18, 2021 | education

In the MNKP trapezoid, Angle M = 45 degrees, Angle P = 30 degrees, the sides are 8 cm and 10 cm, and the smaller base is 5 cm. Find the middle line of the trapezoid.

To find the midline of a trapezoid, you need to know the lengths of both bases of the trapezoid. The middle line of the trapezoid is equal to half the sum of the bases.

Let us lower two heights from the obtuse angles of the trapezoid N and K to the base of the MR, let them be the heights NH and KE.

Consider the triangle MNH
It is a right-angled triangle (NH is perpendicular to the base of the MP).
Let us express the cosine of the angle M: ​​cosM = MH / MN.
Since the angle M is 45 °, the cosine of 45 ° is √2 / 2, and MN = 8 cm, the equation is: √2 / 2 = MH / 8.
Hence MH = 8√2 / 2 = 4√2 (cm).
Consider the triangle KEP
It is a right-angled triangle (KE is perpendicular to the base of the MP).
Let us express the cosine of the angle P: cosP = EP / KP.
Since the angle P is 30 °, the cosine of 30 ° is √3 / 2, and KP = 10 cm, we get the equation: √3 / 2 = EP / 10.
Hence EP = 10√3 / 2 = 5√3 (cm).
Consider a quadrangle HNKE: NK is parallel to HE (since NK is parallel to MP by the property of the trapezoid bases); NH is parallel to KE (two perpendiculars to one side); angles H and E are equal to 90 °. This means that the quadrilateral HNKE is a rectangle. Therefore, HE = NK = 5 cm.

The base of the MP trapezoid consists of three segments – MH, EH and PE.

Hence, MP = 5 + 5√3 + 4√2 (cm).

Calculate the middle line of the trapezoid
The length of the middle line of the MNKP trapezoid is equal to half the sum of the bases of the trapezoid: (NK + MP) / 2 = (5 + 5√3 + 4√2 + 5) / 2 = (10 + 5√3 + 4√2) / 2 = 5 + 2.5√3 + 2√2 (cm).

Answer: the middle line is 5 + 2.5√3 + 2√2 (cm).



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