Final answer:
Using the distance formula derived from the Pythagorean theorem, the lengths of the sides of triangle XYZ are found to be 9.0, 5.0, and approximately 10.3 units. Angle Y is a right angle (90 degrees), while angles X and Z are determined using trigonometric functions, specifically the inverse tangent function.
Step-by-step explanation:
To find the side lengths of triangle XYZ, we will use the distance formula which is derived from the Pythagorean theorem. This formula helps us calculate the distance between two points in a coordinate plane. The distance formula is √((x_2-x_1)^2+(y_2-y_1)^2), where (x_1, y_1) and (x_2, y_2) are the coordinates of two points.
Applying the distance formula, we can calculate the side lengths of triangle XYZ as follows:
- For side XY, the distance between X(-4, -3) and Y(5, -3) is √((5-(-4))^2+((-3)-(-3))^2) = √(9^2+0^2) = √81 = 9.
- For side XZ, the distance between X(-4, -3) and Z(-4, 2) is √((-4-(-4))^2+((2)-(-3))^2) = √(0^2+5^2) = √25 = 5.
- For side YZ, the distance between Y(5, -3) and Z(-4, 2) is √((5-(-4))^2+((2)-(-3))^2) = √(9^2+5^2) = √106 = approximately 10.3.
To find the measures of each angle, we will use trigonometry. Since triangle XYZ has both horizontal and vertical sides, we know that angle Y is a right angle (90 degrees). We can use the tangent function to find angles X and Z. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. Therefore, for angle X, we have tan(X) = opposite / adjacent = YZ / XZ = 10.3 / 5. Angle X can be found by taking the inverse tangent (tan^-1) of that ratio. For angle Z, we can use the symmetry of the triangle, or calculate tan(Z) = opposite / adjacent = YZ / XY, and then take the inverse tangent.
Finally, these calculations provide the side lengths and the measures of the angles in Triangle XYZ to the nearest tenth and degree, respectively.