Write a coordinate proof to show that triangle FGH congruent triangle FDC
To prove that triangle FGH is congruent to triangle FDC using coordinate geometry, we will compare the corresponding sides and angles of the two triangles.
Given:
- Triangle FGH with vertices F(x1, y1), G(x2, y2), and H(x3, y3)
- Triangle FDC with vertices F(x1, y1), D(x4, y4), and C(x5, y5)
To show triangle FGH ≅ triangle FDC, we need to prove the following conditions:
Condition 1: The lengths of corresponding sides FG and FD are equal.
To calculate the length of a line segment between two points, we use the distance formula:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
We can calculate the lengths of sides FG and FD as follows:
Length of FG = √((x2 - x1)^2 + (y2 - y1)^2)
Length of FD = √((x4 - x1)^2 + (y4 - y1)^2)
If Length of FG = Length of FD, then condition 1 is satisfied.
Condition 2: The lengths of corresponding sides GH and DC are equal.
We can calculate the lengths of sides GH and DC as follows:
Length of GH = √((x3 - x2)^2 + (y3 - y2)^2)
Length of DC = √((x5 - x4)^2 + (y5 - y4)^2)
If Length of GH = Length of DC, then condition 2 is satisfied.
Condition 3: The measure of angle F is equal to the measure of angle F.
For this condition, we can use the concept of slopes. The slope of a line passing through two points (x1, y1) and (x2, y2) is given by:
Slope = (y2 - y1) / (x2 - x1)
We can calculate the slopes of lines FG and FD as follows:
Slope of FG = (y2 - y1) / (x2 - x1)
Slope of FD = (y4 - y1) / (x4 - x1)
If Slope of FG = Slope of FD, then the lines FG and FD are parallel, and condition 3 is satisfied.
If all three conditions are satisfied, we can conclude that triangle FGH is congruent to triangle FDC.
Note: This proof assumes that the given coordinates and triangles are valid and consistent.
To prove that triangle FGH is congruent to triangle FDC using a coordinate proof, we need to show that their corresponding sides and angles are equal.
Let's assign coordinates to the vertices of the triangles to better visualize them. We'll assume that point F is the origin (0,0) and point G is located at (x1, y1) while point H is located at (x2, y2). Additionally, point D is located at (x3, y3) and point C is located at (x4, y4).
Now, let's break down the steps to prove the congruence:
Step 1: Show that the corresponding sides are equal:
To show that the corresponding sides of the triangles are equal, we need to calculate the lengths of the sides and compare them.
Side FG is equal to the distance between points F and G, which can be calculated using the distance formula:
FG = √((x1 - 0)^2 + (y1 - 0)^2)
Side GH is equal to the distance between points G and H, which can also be calculated using the distance formula:
GH = √((x2 - x1)^2 + (y2 - y1)^2)
Side FH is equal to the distance between points F and H, which can be calculated using the distance formula as well:
FH = √((x2 - 0)^2 + (y2 - 0)^2)
Similarly, we can calculate the lengths of sides DC and CD using the distance formula:
DC = √((x4 - x3)^2 + (y4 - y3)^2)
CD = √((x3 - x4)^2 + (y3 - y4)^2)
If FG = CD, GH = DC, and FH = CD, then we can conclude that triangle FGH is congruent to triangle FDC.
Step 2: Show that the corresponding angles are equal:
To show that the corresponding angles are equal, we can calculate the slopes of the sides in each triangle.
The slope of side FG can be calculated as:
m(FG) = (y1 - 0) / (x1 - 0)
The slope of side GH can be calculated as:
m(GH) = (y2 - y1) / (x2 - x1)
The slope of side FH can be calculated as:
m(FH) = (y2 - 0) / (x2 - 0)
Similarly, we can calculate the slopes of sides DC and CD:
m(DC) = (y4 - y3) / (x4 - x3)
m(CD) = (y3 - y4) / (x3 - x4)
If the slopes of FG and CD are equal, GH and DC are equal, and FH and CD are equal, then we can conclude that the corresponding angles are equal.
By demonstrating that the corresponding sides and angles are equal, we successfully prove that triangle FGH is congruent to triangle FDC.