The angle between a diagonal and the longer base of the isosceles trapezoid MNFD is 45°. NK is an altitude to the longer base. If MD = 9 and NF = 5, find:
- NK
- MK
- MF
- [tex]A_{AMNFD}[/tex]
Answers
Answer:
[tex]NK \approx 7.001[/tex], [tex]MK = 2[/tex], [tex]MF \approx 7.001[/tex], [tex]A_{AMNFD} = 49.007[/tex]
Step-by-step explanation:
According to the statement, we find the following inputs:
[tex]\angle MDK = \angle DMF = 45^{\circ}[/tex] (Due to the condition of isosceles trapezoid)
[tex]MD = 9[/tex]
[tex]NF = 5[/tex]
Given than longer base and shorter base are parallel to each other, we conclude that:
[tex]\angle KND = 180^{\circ} -\angle NKD - \angle KDN[/tex]
[tex]\angle KND = 180^{\circ}-90^{\circ}-45^{\circ}[/tex]
[tex]\angle KND = 45^{\circ}[/tex]
[tex]\angle FND = 90^{\circ}-\angle KND[/tex]
[tex]\angle FND = 45^{\circ}[/tex] (By definition of complementary angles)
[tex]\angle FND = \angle NFM = 45^{\circ}[/tex] (Due to the condition of isosceles trapezoid)
[tex]\angle MDK = \angle DMF = \angle FND = \angle NFM = 45^{\circ}[/tex]
[tex]\angle NOF = \angle MOD = \angle MOF = \angle FOD = 90^{\circ}[/tex] (By definitions of complementary and vertical angles and the theorem that states that sum of internal angles within a triangle equals 180º)
[tex]MO = DO = \frac{\sqrt{2}}{2}\cdot MD[/tex] (By theorem for 45-45-90 Right Triangle)
[tex]NO = OF = \frac{\sqrt{2}}{2}\cdot NF[/tex] (By theorem for 45-45-90 Right Triangle)
If we know that [tex]MD = 9[/tex] and [tex]NF = 5[/tex], then we find that:
[tex]DO = MO \approx 6.364[/tex]
[tex]NO = OF \approx 3.536[/tex]
The value of MK is obtained from the following relationship:
[tex]MK = \frac{MD-NF}{2}[/tex]
[tex]MK = \frac{9-5}{2}[/tex]
[tex]MK = 2[/tex]
And the value of KD is calculated from this expression:
[tex]KD = MD-MK[/tex]
[tex]KD = 9-2[/tex]
[tex]KD = 7[/tex]
Now by the Pythagorean Theorem we find that:
[tex]NK = \sqrt{(NO+DO)^{2}-KD^{2}}[/tex]
[tex]NK = \sqrt{9.9^{2}-7^{2}}[/tex]
[tex]NK \approx 7.001[/tex]
And considering the symmetry characteristics of an isosceles trapezoid, we determine MF:
[tex]MF = NO + DO \approx 7.001[/tex]
Lastly, the area of the isosceles trapezoid is determined by the following formula:
[tex]A_{AMNFD} = NF\cdot NK + MK\cdot NK[/tex]
[tex]A_{AMNFD} = NK\cdot (NF+MK)[/tex]
If we know that [tex]NK \approx 7.001[/tex], [tex]NF = 5[/tex] and [tex]MK = 2[/tex], then the area of the figure is:
[tex]A_{AMNFD} = (7.001)\cdot (5+2)[/tex]
[tex]A_{AMNFD} = 49.007[/tex]
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