Mathematics
Grade 10
15 min
SSS Theorem in the coordinate plane
SSS Theorem in the coordinate plane
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1
Introduction & Learning Objectives
Learning Objectives
State the Side-Side-Side (SSS) Congruence Theorem.
Calculate the length of a line segment in the coordinate plane using the Distance Formula.
Identify corresponding sides of two triangles given their vertices as coordinate pairs.
Apply the Distance Formula to all three pairs of corresponding sides to gather evidence for an SSS proof.
Construct a logical argument to prove two triangles in the coordinate plane are congruent using the SSS Theorem.
Write a formal congruence statement for two triangles, ensuring vertices are in corresponding order.
Ever wondered how a GPS knows two triangular routes are the exact same length, or how an architect ensures two support beams are identical? 🗺️ Let's find out how coordinates can prove it!
In this tutorial, we...
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Key Concepts & Vocabulary
TermDefinitionExample
Coordinate PlaneA two-dimensional plane formed by the intersection of a horizontal x-axis and a vertical y-axis. Locations are described by ordered pairs (x, y).The point P(4, -2) is located 4 units to the right of the origin and 2 units down.
Vertices of a TriangleThe points where the sides of a triangle meet. In the coordinate plane, vertices are represented by ordered pairs.For ΔABC, the vertices might be A(1, 5), B(4, 2), and C(-2, 0).
Side-Side-Side (SSS) Congruence TheoremA theorem stating that if three sides of one triangle are congruent (equal in length) to the three corresponding sides of another triangle, then the two triangles are congruent.If AB = DE, BC = EF, and AC = DF, then we can conclude that ΔABC ≅ ΔDEF.
Corresponding SidesSides that are in the sam...
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Core Formulas
The Distance Formula
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
Use this formula to calculate the length of each side of a triangle when you know the coordinates of its vertices. To check for SSS congruence, you must apply this formula to find the lengths of all six sides of the two triangles.
SSS Congruence Condition
If AB = DE, BC = EF, and AC = DF, then \triangle ABC \cong \triangle DEF.
This is the core logical test. After using the Distance Formula to find all six side lengths, you must match up the corresponding sides. If you find three pairs of sides with equal lengths, the triangles are congruent by SSS.
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Challenging
\triangle ABC has vertices A(1, 2), B(1, 5), C(3, 2). \triangle PQR has vertices P(-1, 0), Q(-1, 3), R(x, 0). For what value of x is \triangle ABC \cong \triangle PQR by SSS?
A.3
B.-3
C.1
D.-2
Challenging
\triangle DEF has vertices D(0, 0), E(4, 3), and F(4, 0). If \triangle GHI is congruent to \triangle DEF by SSS, what type of triangle is \triangle GHI?
A.right scalene triangle
B.An isosceles triangle
C.An equilateral triangle
D.An obtuse scalene triangle
Challenging
Given \triangle ABC with vertices A(2, 2), B(5, 6), and C(5, 2). Which of the triangles defined by the vertex sets below is congruent to \triangle ABC by SSS?
A.D(0,0), E(3,5), F(3,0)
B.D(-1,-1), E(2,4), F(2,-1)
C.D(1,1), E(4,4), F(4,1)
D.D(-2,-2), E(-5,-6), F(-5,-2)
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