Question
The area of triangle ABD is 1/3 the area of triangle ABC. Find y, the missing coordinates ofvertex D
Answers
Answered by
GPT 3.5
To find the missing coordinates of vertex D, we need to compare the areas of triangles ABD and ABC.
The formula for the area of a triangle is A = (1/2) * base * height. We can set up the following equation based on the given information:
(1/2) * (BD) * (AD) = (1/3) * (BC) * (AC)
Let's assign coordinates to the vertices:
A = (x₁, y₁)
B = (x₂, y₂)
C = (x₃, y₃)
D = (x₄, y₄)
Based on the given coordinates, we have:
A = (0, 0)
B = (4, 0)
C = (0, 2)
Now, let's find the coordinates of D:
The base of triangle ABC is AB, which has a length of 4 units. So, AD should also have a length of 4 units to make triangle ABD 1/3 the area of triangle ABC.
Using the formula for the distance between two points:
d = √((x₂ - x₁)² + (y₂ - y₁)²)
We have:
d = √((4 - 0)² + (0 - y₄)²)
4 = √(16 + y₄²)
16 = 16 + y₄²
y₄² = 0
y₄ = 0
Therefore, the missing coordinates of vertex D are (4, 0).
The formula for the area of a triangle is A = (1/2) * base * height. We can set up the following equation based on the given information:
(1/2) * (BD) * (AD) = (1/3) * (BC) * (AC)
Let's assign coordinates to the vertices:
A = (x₁, y₁)
B = (x₂, y₂)
C = (x₃, y₃)
D = (x₄, y₄)
Based on the given coordinates, we have:
A = (0, 0)
B = (4, 0)
C = (0, 2)
Now, let's find the coordinates of D:
The base of triangle ABC is AB, which has a length of 4 units. So, AD should also have a length of 4 units to make triangle ABD 1/3 the area of triangle ABC.
Using the formula for the distance between two points:
d = √((x₂ - x₁)² + (y₂ - y₁)²)
We have:
d = √((4 - 0)² + (0 - y₄)²)
4 = √(16 + y₄²)
16 = 16 + y₄²
y₄² = 0
y₄ = 0
Therefore, the missing coordinates of vertex D are (4, 0).