Find the lengths of the sides of the triangle PQR. Is it a right triangle? Is it an isosceles triangle? P(2, -1, 0), Q(4, 1, 1), R(4, -5, 4)

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oppturf · Accepted Answer

Step 1 Distance formula The distance between two points P1=(x1,y1,z1) and P2=(x2,y2,z2), denoted by d(P1,P2) is d(P1,P2)=(x2−x1)2+(y2−y1)2+(z2−z1)2 The length of side with vertices P(2,-1,0) and Q(4,1,1) is PQ=(2−4)2+(−1−1)2+(0−1)2 PQ=(−2)2+(−2)2+(−1)2 PQ=4+4+1=9=3 The length of side with vertices P(2,-1,0) and R(4,-5,1)4 is PR=(2−4)2+(−1−(−5))2+(0−4)2 PR=(−2)2+(4)2+(−4)2 PR=4+16+16=36=6 The length of side with vertices Q(2,-1,0) and R(4,-5,1)4 is QR=(4−4)2+(1−(−5))2+(1−4)2 QR=(0)2+(6)2+(−3)2 QR=62+32 Step 2 Since QR2=PQ2+PR2=62+32 This is a right triangle Result This is a right triangle

RizerMix · Accepted Answer

Answer:- Lengths of the sides: PQ=3, QR=35, and RP=6.- Right triangle: No.- Isosceles triangle: Yes.Explanation:d=(x2−x1)2+(y2−y1)2+(z2−z1)2Let&#039;s calculate the lengths of the sides PQ, QR, and RP.Using the distance formula, we have:Length PQ=(4−2)2+(1−(−1))2+(1−0)2=22+22+12=4+4+1=9=3.Length QR=(4−4)2+(−5−1)2+(4−1)2=02+(−6)2+32=0+36+9=45=35.Length RP=(4−2)2+(−5−(−1))2+(4−0)2=22+(−4)2+42=4+16+16=36=6.Therefore, the lengths of the sides of triangle PQR are PQ=3, QR=35, and RP=6.To determine if triangle PQR is a right triangle, we can check if any of the sides satisfy the Pythagorean theorem. According to the theorem, in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.Let&#039;s check if PQ, QR, and RP satisfy the Pythagorean theorem:PQ2+QR2=32+(35)2=9+9·5=9+45=54≠RP2=62=36Since PQ2+QR2≠RP2, triangle PQR is not a right triangle.To determine if triangle PQR is an isosceles triangle, we need to check if any two sides have the same length.Comparing the lengths, we can see that PQ=QR, so triangle PQR is an isosceles triangle.

Vasquez · Accepted Answer

To find the lengths of the sides of triangle PQR, we can use the distance formula in three-dimensional space. The distance d between two points (x1,y1,z1) and (x2,y2,z2) is given by:d=(x2−x1)2+(y2−y1)2+(z2−z1)2Using this formula, we can calculate the lengths of the sides PQ, QR, and RP.Let&#039;s calculate the lengths of the sides:Side PQ:PQ=(4−2)2+(1−(−1))2+(1−0)2=22+22+12=4+4+1=9=3Side QR:QR=(4−4)2+(−5−1)2+(4−1)2=02+(−6)2+32=0+36+9=45=35Side RP:RP=(4−2)2+(−5−(−1))2+(4−0)2=22+(−4)2+42=4+16+16=36=6To determine if triangle PQR is a right triangle, we can check if any of the sides satisfy the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.Let&#039;s check:PQ2+QR2=32+(35)2=9+45=54RP2=62=36Since RP2≠PQ2+QR2, triangle PQR is not a right triangle.To determine if triangle PQR is an isosceles triangle, we can compare the lengths of its sides. An isosceles triangle has at least two sides of equal length.Comparing the lengths of the sides:PQ=3QR=35RP=6Since none of the sides have the same length, triangle PQR is not an isosceles triangle.

Don Sumner · Accepted Answer

The distance between two points P(x1,y1,z1) and Q(x2,y2,z2) is given by:d=(x2−x1)2+(y2−y1)2+(z2−z1)2Let&#039;s calculate the lengths of the sides PQ, QR, and RP and check if any sides are equal in length to determine if it is an isosceles triangle.Given the points:P(2,−1,0),Q(4,1,1),R(4,−5,4).First, let&#039;s calculate the length of side PQ:PQ=(4−2)2+(1−(−1))2+(1−0)2=22+22+12=9=3.Now, let&#039;s calculate the length of side QR:QR=(4−4)2+(−5−1)2+(4−1)2=02+(−6)2+32=45=35.Lastly, let&#039;s calculate the length of side RP:RP=(2−4)2+(−1−(−5))2+(0−4)2=(−2)2+42+(−4)2=36=6.Hence, the lengths of the sides of triangle PQR are:PQ=3,QR=35,RP=6.To determine if it is a right triangle, we need to check if any of the sides satisfy the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.Let&#039;s check if the triangle PQR is a right triangle by using the Pythagorean theorem.PQ2+QR2=32+(35)2=9+45=54.RP2=62=36.Since PQ2+QR2≠RP2, the triangle PQR is not a right triangle.Now, let&#039;s check if it is an isosceles triangle. An isosceles triangle has at least two sides of equal length.From the lengths we calculated earlier:PQ=3,QR=35,RP=6.We can see that no two sides are equal in length. Therefore, the triangle PQR is not an isosceles triangle.In summary, the lengths of the sides of triangle PQR are:PQ=3,QR=35,RP=6.It is not a right triangle and not an isosceles triangle.

Find the lengths of the sides of the triangle PQR. Is it a right triangle? Is it

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