Find distances, section points, and areas of triangles using coordinates on the Cartesian plane.
Distance = √[(x₂-x₁)²+(y₂-y₁)²]. Section formula gives the point dividing a segment in ratio m:n.
Find the distance between (3,4) and (7,1).
√[(7-3)²+(1-4)²]=√[16+9]=5
Find the point dividing (1,2) and (7,8) in the ratio 1:2.
x=(1×7+2×1)/3=3; y=(1×8+2×2)/3=4; (3,4)
Find the distance between (4,5) and (9,17).
√[(9-4)²+(17-5)²]=√[25+144]=13
Area = ½|x₁(y₂-y₃)+x₂(y₃-y₁)+x₃(y₁-y₂)|, using the coordinates of the three vertices.
Find the area of a triangle with vertices (0,0), (4,0), (0,3).
½|0(0-3)+4(3-0)+0(0-0)|=½×12=6
Find the area of a triangle with vertices (1,1), (4,1), (4,5).
½|1(1-5)+4(5-1)+4(1-1)|=½|-4+16+0|=6
Find the area of a triangle with vertices (1,2), (5,2), (5,6).
½|1(2-6)+5(6-2)+5(2-2)|=½|-4+20+0|=8
1. Find the distance between (2,3) and (5,7).
2. Find the point dividing (0,0) and (10,15) in the ratio 2:3.
3. Find the area of a triangle with vertices (0,0), (6,0), (0,4).
4. Find the area of a triangle with vertices (2,3), (5,3), (5,7).
5. Find the midpoint of (2,-3) and (8,5).
6. Find the distance between (3,4) and (8,16).
7. Find the point dividing (2,4) and (14,20) in the ratio 1:3.
8. Find the area of a triangle with vertices (0,0), (8,0), (0,6).
9. Find the area of a triangle with vertices (3,4), (7,4), (7,9).
10. Find the midpoint of (5,-2) and (11,8).