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)

Question
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)

Answers (1)

2021-02-03
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(-3,1), B(2,-1), C(6,9)
\(\displaystyle{c}^{{2}}={\left|{A}{B}\right|}^{{2}}={\left({2}+{3}\right)}^{{2}}+{\left(-{1}-{1}\right)}^{{2}}={25}+{4}={29}\)
\(\displaystyle{a}^{{2}}={\left|{B}{C}\right|}^{{2}}={\left({6}-{2}\right)}^{{2}}+{\left({9}+{1}\right)}^{{2}}={16}+{100}={116}\)
\(\displaystyle{b}^{{2}}={\left|{C}{A}\right|}^{{2}}={\left({6}+{3}\right)}^{{2}}+{\left({9}-{1}\right)}^{{2}}={81}+{64}={145}\)
\(\displaystyle{a}^{{2}}\ne{b}^{{2}}+{c}^{{2}}\)
145=116+29
0

Relevant Questions

asked 2021-02-03
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)
asked 2020-12-29
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)
asked 2021-01-15
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)
asked 2021-03-02
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)
asked 2021-05-18
The student engineer of a campus radio station wishes to verify the effectivencess of the lightning rod on the antenna mast. The unknown resistance \(\displaystyle{R}_{{x}}\) is between points C and E. Point E is a "true ground", but is inaccessible for direct measurement because the stratum in which it is located is several meters below Earth's surface. Two identical rods are driven into the ground at A and B, introducing an unknown resistance \(\displaystyle{R}_{{y}}\). The procedure for finding the unknown resistance \(\displaystyle{R}_{{x}}\) is as follows. Measure resistance \(\displaystyle{R}_{{1}}\) between points A and B. Then connect A and B with a heavy conducting wire and measure resistance \(\displaystyle{R}_{{2}}\) between points A and C.Derive a formula for \(\displaystyle{R}_{{x}}\) in terms of the observable resistances \(\displaystyle{R}_{{1}}\) and \(\displaystyle{R}_{{2}}\). A satisfactory ground resistance would be \(\displaystyle{R}_{{x}}{<}{2.0}\) Ohms. Is the grounding of the station adequate if measurments give \(\displaystyle{R}_{{1}}={13}{O}{h}{m}{s}\) and R_2=6.0 Ohms?
asked 2021-05-03
A charge of \(\displaystyle{6.00}\times{10}^{{-{9}}}\) C and a charge of \(\displaystyle-{3.00}\times{10}^{{-{9}}}\) C are separated by a distance of 60.0 cm. Find the position at which a third charge, of \(\displaystyle{12.0}\times{10}^{{-{9}}}\) C, can be placed so that the net electrostatic force on it is zero.
asked 2021-02-10
Two light sources of identical strength are placed 8 m apart. An object is to be placed at a point P on a line ? parallel to the line joining the light sources and at a distance d meters from it (see the figure). We want to locate P on ? so that the intensity of illumination is minimized. We need to use the fact that the intensity of illumination for a single source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source.
asked 2021-03-29
Two stationary point charges +3 nC and + 2nC are separated bya distance of 50cm. An electron is released from rest at a pointmidway between the two charges and moves along the line connectingthe two charges. What is the speed of the electron when it is 10cmfrom +3nC charge?
Besides the hints I'd like to ask you to give me numericalsolution so I can verify my answer later on. It would be nice ifyou could write it out, but a numerical anser would be fine alongwith the hint how to get there.
asked 2021-03-06
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. \(\displaystyle{\cos{\theta}}=\frac{{21}}{{5}}\)
asked 2021-04-11
The equation F=−vex(dm/dt) for the thrust on a rocket, can also be applied to an airplane propeller. In fact, there are two contributions to the thrust: one positive and one negative. The positive contribution comes from air pushed backward, away from the propeller (so dm/dt<0), at a speed vex relative to the propeller. The negative contribution comes from this same quantity of air flowing into the front of the propeller (so dm/dt>0) at speed v, equal to the speed of the airplane through the air.
For a Cessna 182 (a single-engine airplane) flying at 130 km/h, 150 kg of air flows through the propeller each second and the propeller develops a net thrust of 1300 N. Determine the speed increase (in km/h) that the propeller imparts to the air.
...