# The converse of the Pythagorean theorem is also a true statement: If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle. Use the distance formula and the Pythagorean theorem to determine whether the set of points could be vertices of a right triangle. (-4,3), (0,5), and (3,-4) Question
Right triangles and trigonometry The converse of the Pythagorean theorem is also a true statement: If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle. Use the distance formula and the Pythagorean theorem to determine whether the set of points could be vertices of a right triangle.
(-4,3), (0,5), and (3,-4) 2021-03-03
We know that the square of distance between two points $$\displaystyle{\left({x}_{{1}},{y}_{{1}}\right)}{\quad\text{and}\quad}{\left({x}_{{2}},{y}_{{2}}\right)}{i}{s}{\left({x}_{{2}}-{x}_{{1}}\right)}^{{2}}+{\left(_{y}{2}-{y}_{{1}}\right)}^{{2}}$$ and the converse of pythagorus theorem , we have if sum of square of any two side is equal to square of third side of triangle, then it is right angled triangle.
Using distance formula and converse of pythagorus theorem , we find that ABC is right angled triangle right angled at A
A(-4,3), B(0,5), C(3,-4)
$$\displaystyle{c}^{{2}}={\left|{A}{B}\right|}^{{2}}={\left({0}+{4}\right)}^{{2}}+{\left({5}-{3}\right)}^{{2}}={16}+{4}={20}$$
$$\displaystyle{a}^{{2}}={\left|{B}{C}\right|}^{{2}}={\left({3}-{0}\right)}^{{2}}+{\left(-{4}-{5}\right)}^{{2}}={9}+{81}={90}$$
$$\displaystyle{b}^{{2}}={\left|{C}{A}\right|}^{{2}}={\left({3}+{4}\right)}^{{2}}+{\left(-{4}-{3}\right)}^{{2}}={49}+{49}={98}$$
$$\displaystyle{a}^{{2}}\ne{b}^{{2}}+{c}^{{2}}$$
$$\displaystyle{98}\ne{90}+{20}$$

### Relevant Questions The converse of the Pythagorean theorem is also a true statement: If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle. Use the distance formula and the Pythagorean theorem to determine whether the set of points could be vertices of a right triangle.
(-4,3), (0,5), and (3,-4) The converse of the Pythagorean theorem is also a true statement: If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle. Use the distance formula and the Pythagorean theorem to determine whether the set of points could be vertices of a right triangle.
(-3,1), (2,-1), and (6,9) The converse of the Pythagorean theorem is also a true statement: If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle. Use the distance formula and the Pythagorean theorem to determine whether the set of points could be vertices of a right triangle.
(-3,1), (2,-1), and (6,9) The converse of the Pythagorean theorem is also a true statement: If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle. Use the distance formula and the Pythagorean theorem to determine whether the set of points could be vertices of a right triangle.
(-3,1), (2,-1), and (6,9) Sketch a right triangle corresponding to the trigonometric function of the acute angle theta. Use the Pythagorean Theorem to determine the third side and then find the other five trigonometric functions of theta. $$\displaystyle{\cos{\theta}}=\frac{{21}}{{5}}$$ Use function notation to describe the way the second variable (DV) depends upon the first variable (IV). Determine the domain and range for each, determine if there is a positive, negative, or no relationship, and explain your answers.
A)IV: an acute angle V in a right triangle: DV: the area B of the triangle if the hypotenuse is a fixed length G.
B)IV: one leg P of a right triangle: DV: the hypotenuse G of the right triangle if the other leg is 2
C)IV: the hypotenuse G of a right triangle: DV: the other leg P of the right triangle is one leg is 5.   If $$\displaystyle{8}{R}^{{2}}={a}^{{2}}+{b}^{{2}}+{c}^{{2}}$$ ,prove that the triangle is right angle triangle. Here a,b and c are the lengths of side of triangle and R is circum radius. 1. Find each of the requested values for a population with a mean of $$? = 40$$, and a standard deviation of $$? = 8$$ A. What is the z-score corresponding to $$X = 52?$$ B. What is the X value corresponding to $$z = - 0.50?$$ C. If all of the scores in the population are transformed into z-scores, what will be the values for the mean and standard deviation for the complete set of z-scores? D. What is the z-score corresponding to a sample mean of $$M=42$$ for a sample of $$n = 4$$ scores? E. What is the z-scores corresponding to a sample mean of $$M= 42$$ for a sample of $$n = 6$$ scores? 2. True or false: a. All normal distributions are symmetrical b. All normal distributions have a mean of 1.0 c. All normal distributions have a standard deviation of 1.0 d. The total area under the curve of all normal distributions is equal to 1 3. Interpret the location, direction, and distance (near or far) of the following zscores: $$a. -2.00 b. 1.25 c. 3.50 d. -0.34$$ 4. You are part of a trivia team and have tracked your team’s performance since you started playing, so you know that your scores are normally distributed with $$\mu = 78$$ and $$\sigma = 12$$. Recently, a new person joined the team, and you think the scores have gotten better. Use hypothesis testing to see if the average score has improved based on the following 8 weeks’ worth of score data: $$82, 74, 62, 68, 79, 94, 90, 81, 80$$. 5. You get hired as a server at a local restaurant, and the manager tells you that servers’ tips are $42 on average but vary about $$12 (\mu = 42, \sigma = 12)$$. You decide to track your tips to see if you make a different amount, but because this is your first job as a server, you don’t know if you will make more or less in tips. After working 16 shifts, you find that your average nightly amount is$44.50 from tips. Test for a difference between this value and the population mean at the $$\alpha = 0.05$$ level of significance.