Question

you do: check for understanding
what are the coordinates of p if pqr is a right triangle and q(4, -1) and r(8, 3)?
a. p(8, -5)
b. p(12, -1)
c. p(0, -5)
d. p(6, 7)

Explanation:

Step1: Recall right triangle condition

For a right triangle, the dot product of two perpendicular sides is 0. We test each option by checking if vectors from the right angle point are perpendicular.

Step2: Test Option A: P(8, -5)

Vectors: $\overrightarrow{QP}=(8-4, -5-(-1))=(4, -4)$, $\overrightarrow{QR}=(8-4, 3-(-1))=(4, 4)$
Dot product: $4\times4 + (-4)\times4 = 16 - 16 = 0$
This confirms $\overrightarrow{QP} \perp \overrightarrow{QR}$, so $\triangle PQR$ is right-angled at Q.

Step3: Verify other options (optional)

• Option B: P(12, -1): $\overrightarrow{QP}=(8,0)$, $\overrightarrow{QR}=(4,4)$, dot product $8\times4 + 0\times4=32

eq0$; $\overrightarrow{RQ}=(-4,-4)$, $\overrightarrow{RP}=(4,-4)$, dot product $(-4)\times4 + (-4)\times(-4)=-16+16=0$? Wait no, $\overrightarrow{RP}=(12-8,-1-3)=(4,-4)$, $\overrightarrow{RQ}=(4-8,-1-3)=(-4,-4)$. Dot product $4\times(-4)+(-4)\times(-4)=-16+16=0$, but wait our first test already found a valid option, but let's confirm: Wait no, for option A we already have a valid right triangle, but let's check again. Wait no, actually for option A, we have a right angle, so it is a valid answer. But let's confirm:
Wait for option A: $\triangle PQR$ has Q(4,-1), R(8,3), P(8,-5). QP is (4,-4), QR is (4,4), dot product 0, so right angle at Q, which is valid.

Step4: Confirm validity

Since Option A satisfies the right triangle condition, we can conclude it is a correct answer.

Answer:

A. P(8, -5)