Question

Plot each point and form the triangle \(A B C\). Show that the triangle is a right triangle. Find its area. $$ A=(-2,5) ; \quad B=(1,3) ; \quad C=(-1,0) $$

Short answer

Triangle ABC is a right triangle with an area of 6.5 square units.

Step-by-step solution

Plot Points

Plot the given points on the coordinate plane. The points are: A(-2, 5), B(1, 3), C(-1, 0). Mark these points clearly.

Form the Triangle

Connect the points A, B, and C to form triangle ABC.

Calculate the Distance Between Points

Use the distance formula to find the lengths of sides AB, BC, and AC. The distance formula is: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) Find AB: \(AB = \sqrt{(1 - (-2))^2 + (3 - 5)^2}\) \(AB = \sqrt{(1 + 2)^2 + (-2)^2}\) \(AB = \sqrt{3^2 + (-2)^2}\) \(AB = \sqrt{9+4}\) \(AB = \sqrt{13}\) Next, find BC: \(BC = \sqrt{(-1 - 1)^2 + (0 - 3)^2}\) \(BC = \sqrt{(-2)^2 + (-3)^2}\) \(BC = \sqrt{4 + 9}\) \(BC = \sqrt{13}\) Lastly, find AC: \(AC = \sqrt{(-1 - (-2))^2 + (0 - 5)^2}\) \(AC = \sqrt{(-1 + 2)^2 + (-5)^2}\) \(AC = \sqrt{1^2 + 25}\) \(AC = \sqrt{26}\)

Verify Right Triangle

To verify it is a right triangle, use the Pythagorean theorem: \(a^2 + b^2 = c^2\) Since AB and BC are the two shortest sides, and AC is the longest: \((\sqrt{13})^2 + (\sqrt{13})^2 = (\sqrt{26})^2\) \(13 + 13 = 26\) This verifies that triangle ABC is a right triangle.

Calculate the Area

The area of a right triangle can be calculated using: \(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\) Since AB and BC are the perpendicular sides: \(\text{Area} = \frac{1}{2} \times \sqrt{13} \times \sqrt{13} \) \(\text{Area} = \frac{1}{2} \times 13 \) \(\text{Area} = 6.5 \)

Key concepts

coordinate geometry

Coordinate geometry, also called analytic geometry, connects algebra and geometry by utilizing a coordinate system. Each point in coordinate geometry is defined by an ordered pair \(x, y\). These coordinates denote the horizontal distance (x) and vertical distance (y) from a fixed origin point (0,0).

In this exercise, we plot three points A((-2, 5)), B((1, 3)), and C((-1, 0)) on the coordinate plane. Each coordinate pair specifies a unique location, allowing us to visualize relationships, such as the sides of a triangle. By plotting the points and connecting them, we can form triangle ABC.

distance formula

The distance formula is essential in coordinate geometry for calculating the straight-line distance between two points on a plane. This formula derives from the Pythagorean theorem and is written as: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

In our exercise, we use the distance formula to find the lengths of the sides of triangle ABC. Let's go through each calculation:

• AB: \(AB = \sqrt{(1 - (-2))^2 + (3 - 5)^2} \)
• BC: \(BC = \sqrt{(-1 - 1)^2 + (0 - 3)^2} \)
• AC: \(AC = \sqrt{(-1 - (-2))^2 + (0 - 5)^2}\)

After finding these lengths, we can determine the triangle's type and verify its properties.

Pythagorean theorem

The Pythagorean theorem is a fundamental principle in geometry that applies specifically to right triangles. It states: \[ a^2 + b^2 = c^2 \]
The theorem notes that in a right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides.

In our exercise, upon finding the lengths of sides AB and BC are both \(\sqrt{13}\), and AC is \(\sqrt{26}\), we check if: \[ (\sqrt{13})^2 + (\sqrt{13})^2 = (\sqrt{26})^2 \] This simplifies to: \[ 13 + 13 = 26 \] and confirms that triangle ABC is a right triangle. This step is important for validating the triangle's properties and ensuring calculations for the area are accurate.

triangle area calculation

The area of a triangle can be calculated using different methods, but for right triangles, the simplest formula is: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]

In triangle ABC, since AB and BC are perpendicular, they serve as the base and height. Hence, the area calculation is as follows: \[\text{Area} = \frac{1}{2} \times \sqrt{13} \times \sqrt{13} \] This simplifies to: \[\text{Area} = \frac{1}{2} \times 13 = 6.5 \]

By following these steps, you can accurately determine the area of any right triangle, reinforcing the relationship between the triangle's sides and its area.