A right triangle and an isosceles triangle are graphed on the coordinate grid. The shaded section represents all the points located inside both triangles.
Which coordinates represent the location of a point inside the shaded section?
(─4.5, 1.5)
(1.5, ─4.5)
(4.5, ─1.5)
(─1.5, 4.5)
(1.5, ─4.5)
To determine if a point is inside the shaded section of both triangles, we need to determine if it is inside the right triangle and the isosceles triangle.
For the right triangle, we can find the equation of the hypotenuse as well as the equations of the two perpendicular sides. Then, we can check if the point satisfies the inequalities of these equations.
For the isosceles triangle, we need to find the equation for the base and one of the sides. Then, we can again check if the point satisfies the inequalities of these equations.
Let's start with the right triangle:
First, let's find the equation of the hypotenuse. Since it's a right triangle, the hypotenuse can be found using the Pythagorean Theorem.
Letting the coordinates of the two vertices of the hypotenuse be (x1, y1) and (x2, y2), the equation of the hypotenuse can be written as:
(y - y1) / (y2 - y1) = (x - x1) / (x2 - x1)
Using the values from the question, the equation of the hypotenuse is:
(y - 1.5) / (1.5 - 4.5) = (x + 4.5) / (-4.5 - 1.5)
Simplifying this equation, we get:
(y - 1.5) / (-3) = (x + 4.5) / (-6)
Cross multiplying and rearranging, we have:
6(y - 1.5) = -3(x + 4.5)
6y - 9 = -3x - 13.5
6y + 3x = -4.5
Next, let's find the equations of the two perpendicular sides of the right triangle.
One side is the vertical line that passes through the point (-4.5, 1.5). The equation for this line is:
x = -4.5
The other side is the horizontal line that passes through the point (-4.5, 1.5). The equation for this line is:
y = 1.5
Now, let's check if the point (─4.5, 1.5) satisfies the inequalities of the right triangle.
For the hypotenuse, substituting the point (─4.5, 1.5) into the equation 6y + 3x = -4.5, we get:
6(1.5) + 3(-4.5) = -4.5
9 - 13.5 = -4.5
-4.5 = -4.5
Since the left-hand side equals the right-hand side, the point satisfies the equation of the hypotenuse.
Next, substituting the point (─4.5, 1.5) into the equation x = -4.5, we get:
-4.5 = -4.5
The point satisfies the equation for the vertical side as well.
Finally, substituting the point (─4.5, 1.5) into the equation y = 1.5, we get:
1.5 = 1.5
The point satisfies the equation for the horizontal side.
Therefore, the point (─4.5, 1.5) is inside the shaded section of the right triangle.
For the isosceles triangle, we need to find the equation for the base and one of the sides.
Let's assume the coordinates of the two vertices at the base are (x1, y1) and (x2, y2), and the coordinates of the vertex at the top are (x3, y3).
Using the coordinates given in the question, we know that the coordinates of the two vertices at the base are (-4.5, 1.5) and (1.5, -4.5). The coordinates of the vertex at the top are (1.5, -1.5).
To find the equation of the base, we use the midpoint formula:
(x, y) = ( (x1 + x2) / 2, (y1 + y2) / 2 )
Substituting the coordinates of the two vertices at the base into the midpoint formula, we get:
(x, y) = ( (─4.5 + 1.5) / 2, (1.5 + (−4.5)) / 2 )
= (-3 / 2, -3 / 2)
= (-1.5, -1.5)
The equation for the base can be written as:
y = -1.5
Now, let's check if the point (─4.5, 1.5) satisfies the inequality of the isosceles triangle.
Substituting the point (─4.5, 1.5) into the equation y = -1.5, we get:
1.5 = -1.5
The point does not satisfy the equation for the base of the isosceles triangle.
Therefore, the point (─4.5, 1.5) is not inside the shaded section of the isosceles triangle.
To summarize, the coordinates that represent the location of a point inside the shaded section of both triangles are:
(─4.5, 1.5)