Studdy AI Logo
Math  /  Geometry

QuestionPoints P,Q,RP, Q, R and SS have position vectors p=(63),q=(−3−5),r=(1−3)\mathbf{p}=\binom{6}{3}, \mathbf{q}=\binom{-3}{-5}, \mathbf{r}=\binom{1}{-3} and s=(105)\mathbf{s}=\binom{10}{5} Prove that the quadrilateral PQRSP Q R S is a parallelogram.

Studdy Solution

STEP 1

1. The position vectors of points P,Q,R, P, Q, R, and S S are given.
2. A quadrilateral is a parallelogram if opposite sides are equal in length and parallel.

STEP 2

1. Calculate the vectors representing the sides of the quadrilateral.
2. Show that opposite sides are equal in length.
3. Show that opposite sides are parallel.

STEP 3

Calculate the vectors representing the sides of the quadrilateral:
PQundefined=q−p=(−3−5)−(63)=(−3−6−5−3)=(−9−8)\overrightarrow{PQ} = \mathbf{q} - \mathbf{p} = \binom{-3}{-5} - \binom{6}{3} = \binom{-3 - 6}{-5 - 3} = \binom{-9}{-8}
QRundefined=r−q=(1−3)−(−3−5)=(1+3−3+5)=(42)\overrightarrow{QR} = \mathbf{r} - \mathbf{q} = \binom{1}{-3} - \binom{-3}{-5} = \binom{1 + 3}{-3 + 5} = \binom{4}{2}
RSundefined=s−r=(105)−(1−3)=(10−15+3)=(98)\overrightarrow{RS} = \mathbf{s} - \mathbf{r} = \binom{10}{5} - \binom{1}{-3} = \binom{10 - 1}{5 + 3} = \binom{9}{8}
SPundefined=p−s=(63)−(105)=(6−103−5)=(−4−2)\overrightarrow{SP} = \mathbf{p} - \mathbf{s} = \binom{6}{3} - \binom{10}{5} = \binom{6 - 10}{3 - 5} = \binom{-4}{-2}

STEP 4

Show that opposite sides are equal in length:
Calculate the magnitudes:
∣PQundefined∣=(−9)2+(−8)2=81+64=145|\overrightarrow{PQ}| = \sqrt{(-9)^2 + (-8)^2} = \sqrt{81 + 64} = \sqrt{145}
∣RSundefined∣=92+82=81+64=145|\overrightarrow{RS}| = \sqrt{9^2 + 8^2} = \sqrt{81 + 64} = \sqrt{145}
∣QRundefined∣=42+22=16+4=20|\overrightarrow{QR}| = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20}
∣SPundefined∣=(−4)2+(−2)2=16+4=20|\overrightarrow{SP}| = \sqrt{(-4)^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20}
Since ∣PQundefined∣=∣RSundefined∣ |\overrightarrow{PQ}| = |\overrightarrow{RS}| and ∣QRundefined∣=∣SPundefined∣ |\overrightarrow{QR}| = |\overrightarrow{SP}| , opposite sides are equal.

STEP 5

Show that opposite sides are parallel:
Check if vectors are scalar multiples:
PQundefined=(−9−8),RSundefined=(98)\overrightarrow{PQ} = \binom{-9}{-8}, \quad \overrightarrow{RS} = \binom{9}{8}
PQundefined=−1×RSundefined\overrightarrow{PQ} = -1 \times \overrightarrow{RS}
QRundefined=(42),SPundefined=(−4−2)\overrightarrow{QR} = \binom{4}{2}, \quad \overrightarrow{SP} = \binom{-4}{-2}
QRundefined=−1×SPundefined\overrightarrow{QR} = -1 \times \overrightarrow{SP}
Since PQundefined \overrightarrow{PQ} is parallel to RSundefined \overrightarrow{RS} and QRundefined \overrightarrow{QR} is parallel to SPundefined \overrightarrow{SP} , opposite sides are parallel.
Thus, the quadrilateral PQRS PQRS is a parallelogram.

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord