Find the area of a triangle whose vertices have coordinates (0,0) (10,7) (7,10).

First, let’s find the lengths of all three sides of this geometric figure.

Let’s use the formula for the distance between two points A and B on the coordinate plane with coordinates A (x1; y1) and B (x2; y2):

| AB | = √ ((x1 – x2) ² + (y1 – y2) ²).

Find the distance between points with coordinates (0; 0) and (10; 7):

√ ((10 – 0) ² + (7 – 0) ²) = √149.

Find the distance between points with coordinates (0; 0) and (7; 10):

√ ((7 – 0) ² + (10 – 0) ²) = √149.

Find the distance between points with coordinates (7; 10) and (10; 7):

√ ((10 – 7) ² + (7 – 10) ²) = √18 = 3√2.

Find the semi-perimeter of this triangle:

(√149 + √149 + 3√2) / 2 = (2√149 + 3√2) / 2.

Find the area of ​​a triangle using Heron’s formula;

√ ((2√149 + 3√2) / 2) * (3√2 / 2) * (3√2 / 2) * ((2√149 – 3√2) / 2) = (3√2 / 4) * √ (596 – 18) = (3√2 / 4) * √587 = (3√2 / 4) * 17√2 = 51/2 = 25.5.

Answer: 25.5.