Two chords MH and PK are drawn in the circle, they intersect at point E. MH = 14 cm, ME is 2 cm larger than HE.

| May 10, 2021 | education

Two chords MH and PK are drawn in the circle, they intersect at point E. MH = 14 cm, ME is 2 cm larger than HE. Find the area of the triangle PHE if the area of the triangle MEK is 64 cm2.

Determine the lengths of the segments ME and HE, moves MH.

By condition, MH = 14 cm, then HE + ME = 14 cm, and also ME – HE = 2 cm.

Let’s solve a system of two equations.

ME = 2 + HE, then HE + 2 + HE = 14.

2 * HE = 12 cm.

HЕ = 12/2 = 6 cm.

ME = 2 + 6 = 8 cm.

Let us prove the similarity of the triangles РHЕ and MEK.

Angle HEP = KEM as vertical angles, angle HPE = HMK as inscribed angles based on the arc HK. Then the triangles PHE and MEK are similar in two angles.

The coefficient of similarity of triangles is: K = 6/8 = 3/4.

The ratio of the areas of similar triangles is equal to the squared coefficient of their similarity.

Srne / Smek = 9/16.

Srne = 9 * 64/16 = 36 cm2.

Answer: The area of ​​the РHЕ triangle is 35 cm2.



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