In an isosceles triangle ABC, the base BC = 12 cm, the lateral side AB = 10 cm. The ABC triangle is similar to an MRT

In an isosceles triangle ABC, the base BC = 12 cm, the lateral side AB = 10 cm. The ABC triangle is similar to an MRT triangle and angle P = angle A. Find the similarity factor and the area of the MRT triangle if MT = 36 cm.

An isosceles triangle is called a triangle in which two sides are equal and called lateral, and the third unequal – the base.
The similarity coefficient is the number k equal to the ratio of the similar sides of similar triangles:
k = PM / AB = MT / BC = PT / AC.
k = MT / BC;
k = 36/12 = 3.
To calculate the area of ​​a triangle ΔMPT, we use the formula for the area behind two sides and the angle between them:
S = 1/2 a b cos α.
To do this, calculate the side length MP and cos M.
In order to calculate the length of the side MP, you need to multiply the length of the corresponding side of such a triangle by the similarity coefficient:
MP = BA · k;
MP = 10 3 = 30 cm.
To calculate the cosine of the angle ∠M, we draw the height of the PH and use the cosine theorem.
The cosine of an acute angle of a right triangle is the ratio of the adjacent leg to the hypotenuse:
cos M = MH / PM;
Since the height of an isosceles triangle, drawn to the base, divides it in half:
MH = MT / 2;
MH = 36/2 = 18 cm.
cos M = 18/30 = 0.6.
S = 1/2 · 36 · 30 · 0.6 = 324 cm2.
Answer: coefficient of similarity of triangles MPT / ABC = 3; SMPT = 324 cm2.