An isosceles right triangle has two equal length legs and the hypotenuse side as its base
The equation of the sides of the isosceles right triangle are;
x = 0 (equation of the base)
y = 1 - x
y = x + 5
The reason for arriving at the above equations are as follows:
The given parameters of the isosceles right triangle are:
The coordinate of the vertex of the triangle = (-2, 3)
The base of the triangle, x = 0
The required parameter:
The equation of the sides of the isosceles triangle
Strategy:
Define the terms and relate the given information with the given term definitions
Solution:
The base of an isosceles triangle is the side that form the equal base angles with the two equal sides of the triangle
In an isosceles right triangle, the two equal sides of the right triangle are the legs of the right triangle, and therefore, the base of the triangle is the hypotenuse side
The base angles of an isosceles right triangle, [tex]\theta _b[/tex] = 45°
The altitude of the triangle, y = (0 - (-2) = 2
The altitude is a perpendicular bisector of the base, therefore;
[tex]tan(\theta _b) = \dfrac{Altitude}{\dfrac{1}{2} \times Base \ length }[/tex]
Therefore;
[tex]Base \ length = \dfrac{Altitude}{\dfrac{1}{2} \times tan(\theta _b)}[/tex]
We get:
[tex]\mathbf{Base \ length} = \dfrac{2}{\dfrac{1}{2} \times tan(45^{\circ})} \mathbf{= 4~}[/tex]
The coordinate of the midpoint of the base is the horizontal line from the vertex to the base, and therefore, the coordinate of the midpoint of the base is x = 0, and y = 3 = (0, 3)
The base is 4 units long and half the base length = 4/2 = 2 units
The coordinates of vertex at the base is given by adding or subtracting 2 from the y-coordinates of midpoint of the base as follows;
Vertex coordinates at the base = (0, 3 + 2) = (0, 5), and (0, 3 - 2) = (0, 1)
The slope, m₁, of the side, s₁, with endpoints (-2, 3), and (0, 1), is found as follows;
m₁ = (1 - 3)/(0 - (-2)) = -1
Therefore, the equation of the side s₁ is (y - 1) = -1·(x - 0), which gives;
y = -x + 1
The slope, m₂, of the side, s₂, with endpoints (-2, 3), and (0, 5), is found as follows;
m₂ = (5 - 3)/(0 - (-2)) = 1
Therefore, the equation of the other side s₂ is (y - 5) = 1·(x - 0), which gives;
y = x + 5
The equation of the sides are; x = 0 (given), y = 1 - x, and y = x + 5
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