So the ratio of the sides and the hypotenuse, in a 45-45-90 triangle, or a right isosceles triangle. What type of triangle is a 45 45 90 right triangle?.Because the altitude is the longer leg of the 30В°в€’60В°в€’90В° right triangle and its measure is x, the altitude is 10 inches long. In this problem, the length 20 inches represents the longest side in the 30В°в€’60В°в€’90В° right triangle, so 2 xВ = 20, or xВ = 10. The ratio of sides in a 30В°в€’60В°в€’90В° right triangle is x: x : 2 x. Because the perimeter is 60 inches, and the three sides are equal in measure, then each side is 20 inches (60 Г· 3 = 20).

If an altitude is drawn, it creates two 30В°в€’60В°в€’90В° right triangles. The hypotenuse is 16.ДЇigure 7 Using the longer leg of a 30В°в€’60В°в€’90В° triangle to find the hypotenuse.Д®xample 6:В Find the length of an altitude in an equilateral triangle with a perimeter of 60 inches.ДЇigure 8В is an equilateral triangle. x is the longer leg, and it is opposite the 60В°. In Figure 7В , the shorter leg, x, is opposite the 30В°. Consequently, the other two sides must be 4 A and 2(4), or 8.ДЇigure 6 Using the shortest side of a 30В°в€’60В°в€’90В° triangle to find the remaining sides.Д®xample 5:В If the longer leg of a 30В°в€’60В°в€’90В° right triangle is 8, find the length of the hypotenuse. The other two sides are then x (opposite the 60В°) and 2 xВ (opposite the 90В°). The side opposite 90В° is the longest side (hypotenuse) and is 2 or 2 x.Д®xample 4:В If the shortest side of a 30В°в€’60В°в€’90В° right triangle is 4, what is the measure of the other two sides? The side opposite 30В° is the shortest side and is 1 or xВ (Figure 5).ДЇigure 5В The ratios of the sides of a 30В°в€’60В°в€’90В° triangle The ratio of the sides of a 30В°в€’60В°в€’90В° right triangle is 1 : : 2 or xВ : x : 2 xВ placed as follows. A 30В°в€’ 60В°в€’ 90В° right triangleВ has a unique ratio of its sides. Therefore, this is an isosceles right triangle with the ratio of sides xВ : xВ : x Because one leg is 10, the other must also be 10, and the hypotenuse is 10, so yВ = 10 and zВ = 10. Therefore, each side of the square has a measure of 6.Д¤5В° + 90В° + xВ° = 180В° (The sum of the angles of a triangle = 180В°.) xВ = 45В° 6 represents the hypotenuse.Д®xample 3:В What are the measurements of x, y, and zВ in Figure 4В ?ДЇigure 4В Finding the unknown parts of this right triangle So xВ = 6, and each side of the square has a measure of 6. The ratio xВ : xВ : x for isosceles right triangles can now be applied where x = 6. Look at Figure 3В .ДЇigure 3 A diagonal of a square helps create two congruent isosceles right triangles. Method 1: The diagonal of a square divides it into two congruent isosceles right triangles. Using the Pythagorean TheoremВ and the fact that the legs of this right triangle are equal,Д®xample 2:В If the diagonal of a square is 6, find the length of each of its sides. Method 1:В Using the ratio xВ : xВ : x for isosceles right triangles, then xВ = 3, and the other sides must be 3 and 3. The ratio of the sides of an isosceles right triangle is always 1 : 1 : or xВ : x: x (Figure 2В ).ДЇigure 2 The ratios of the sides of an isosceles right triangleД®xample 1:В If one of the equal sides of an isosceles right triangle is 3, what are the measures of the other two sides? (The right angle cannot be one of the equal angles or the sum of the angles would exceed 180В°.) Therefore, in Figure 1В , О” ABCВ is an isosceles right triangle, and the following must always be true. It has two equal sides, two equal angles, and one right angle. An isosceles right triangleВ has the characteristic of both the isosceles and the right triangles.

Summary of Coordinate Geometry Formulas.

Slopes: Parallel and Perpendicular Lines.