The base of the triangle is 26. The medians of its lateral sides are 30 and 39. Find the area of this triangle.

Dan △ ABC: AC = 26, AM = 30 and CK = 39 are the medians drawn to the sides BC and AB, respectively.

1. The length of the median of a triangle is found by the formula:

m = (√ (2 * a² + 2 * b² – c²)) / 2,

where c is the side to which the median is drawn, a and b are the other sides of the triangle.

Then:

AM = (√ (2 * AB² + 2 * AC² – BC²)) / 2;

CK = (√ (2 * AC² + 2 * BC² – AB²)) / 2.

Substitute the data for the value condition:

(√ (2 * AB² + 2 * 26² – BC²)) / 2 = 30;

(√ (2 * 26² + 2 * BC² – AB²)) / 2 = 39.

Let’s denote AB as x, and BC as y and we get a system of equations with two variables:

(√ (2 * x² + 1352 – y²)) / 2 = 30;

(√ (1352 + 2 * y² – x²)) / 2 = 39.

2. Let’s solve the system of equations. In the first equation, we express y:

(√ (2 * x² + 1352 – y²)) / 2 = 30;

√ (2 * x² + 1352 – y²) = 60;

2 * x² + 1352 – y² = 3600;

– y² = 3600 – 2 * x² – 1352;

y² = 2 * x² – 2248;

y = √ (2 * x² – 2248).

Substitute the resulting expression into the second equation:

(√ (1352 + 2 * (2 * x² – 2248) – x²)) / 2 = 39;

√ (1352 + 2 * 2 * x² – 2 * 2248 – x²) = 78;

4 * x² – x² – 4496 + 1352 = 6084;

3 * x² = 6084 + 3144;

3 * x² = 9228;

x² = 9228/3;

x² = 3076;

x = √3076.

Find y:

y = √ (2 * 3076 – 2248) = √ (6152 – 2248) = √3904.

Thus, the sides of △ ABC are equal:

AB = √3076;

BC = √3904;

AC = 26.

3. Find the area △ ABC by Heron’s formula:

S = √ (p * (p – a) * (p – b) * (p – c)),

where p is a semi-perimeter.

Semi-perimeter:

p = (AB + BC + AC) / 2 = (√3076 + √3904 + 26) / 2.

Find the area △ ABC:

S = √ ((√3076 + √3904 + 26) / 2 * ((√3076 + √3904 + 26) / 2 – √3076) * ((√3076 + √3904 + 26) / 2 – √3904) * ((√3076 + √3904 + 26) / 2 – 26)) = √ ((√3076 + √3904 + 26) / 2 * (√3076 + √3904 + 26 – 2 * √3076) / 2 * (√ 3076 + √3904 + 26 – 2 * √3904) / 2 * (√3076 + √3904 + 26 – 52) / 2) = √ ((√3076 + √3904 + 26) / 2 * (- √3076 + √ 3904 + 26) / 2 * (√3076 – √3904 + 26) / 2 * (√3076 + √3904 – 26) / 2) = √ ((3904 + 52√3904 + 676 – 3076) / 4 * (3076 – 676 ​​+ 52√3904 – 3904) / 4) = √ ((52√3904 + 1504) / 4 * (52√3904 – 1504) / 4) = √ ((13√3904 + 376) * (13√3904 – 376)) = √ (659776 – 141376) = √518400 = 720.

Answer: S = 720.