In an isosceles triangle MNK with base MK equal to 10 cm, MN = NK = 20 cm. On the side of NK

| November 29, 2020 | education

In an isosceles triangle MNK with base MK equal to 10 cm, MN = NK = 20 cm. On the side of NK there is point A so that AK: AN = 1: 3. Find AM

In an isosceles triangle MNK with a base MK equal to 10 cm,
MN = NK = 20 cm. Point A lies on the side NK so that AK: AN is 1: 3. Н
AK: KN = 1: 3
Let the coefficient of this ratio be x.
Since NK = 20 = x + 3x = 4x,
AK = 20: 4 = 5cm
Let’s draw AB parallel to the base of MK and AC parallel to the lateral side of NM.
Triangles MNK and ABN are similar with a similarity factor KN: AN = 4: 3
Consequently, MK: AB = 4: 3
10: AB = 4: 3
4AB = 30
AB = 7.5 cm
In an ABMC parallelogram, opposite sides are equal.
BM = AK = AC = 5 cm
MS = 7.5 cm
The ASK triangle is isosceles.
Let us find its height AN by Comrade Pythagoras.
KS = MK-MS = 10-7.5 = 2.5 cm
NK = 1.25 cm
AH² = (AK²-NK²) = (5²-1.25²) = 23.4375
From the right-angled triangle NAM we find AM by Comrade Pythagoras:
AM = √ (MH² + AH²) = √ (7.5² + 23.4375) = √100 = 10 cm



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