Triangle PQR has vertices P(-2,8), Q(2,4), and R(4,6), which best describes triangle PQR?
A- a right triangle with PQ ㅗ QR
B- a right triangle with PQ ㅗ PR
C- an isosceles triangle with PQ congruent to QR
D- an isosceles triangle with PQ congruent with PR
Is B
To determine the best description for triangle PQR, let's analyze the given coordinates:
P(-2,8)
Q(2,4)
R(4,6)
Step 1: Calculate the lengths of the sides using the distance formula (d = √((x2 - x1)^2 + (y2 - y1)^2)):
- Length of PQ:
d(PQ) = √((2 - (-2))^2 + (4 - 8)^2)
= √(4^2 + (-4)^2)
= √(16 + 16)
= √32
= 4√2
- Length of QR:
d(QR) = √((4 - 2)^2 + (6 - 4)^2)
= √(2^2 + 2^2)
= √(4 + 4)
= √8
= 2√2
- Length of PR:
d(PR) = √((-2 - 4)^2 + (8 - 6)^2)
= √((-6)^2 + 2^2)
= √(36 + 4)
= √40
= 2√10
Step 2: Analyze the relationships between the sides and angles:
Looking at the lengths obtained:
- PQ and QR have the same length (4√2 = 2√2 + 2√2), meaning PQ is congruent to QR.
- PR has a different length (2√10), indicating it is not congruent to PQ or QR.
Based on this analysis, we can conclude that the best description for triangle PQR is:
D- an isosceles triangle with PQ congruent to PR.
To determine which description best fits triangle PQR, we need to analyze its sides and angles based on the given coordinates of its vertices.
To start, let's find the lengths of the three sides using the distance formula:
1. Length of side PQ:
PQ = √[(x2 - x1)² + (y2 - y1)²] [Using distance formula]
= √[(2 - (-2))² + (4 - 8)²] [Substituting the coordinates of Q(2,4) and P(-2,8)]
= √[4² + (-4)²]
= √[16 + 16]
= √32
= 4√2
2. Length of side QR:
QR = √[(x2 - x1)² + (y2 - y1)²] [Using distance formula]
= √[(4 - 2)² + (6 - 4)²] [Substituting the coordinates of R(4,6) and Q(2,4)]
= √[2² + 2²]
= √[4 + 4]
= √8
= 2√2
3. Length of side PR:
PR = √[(x2 - x1)² + (y2 - y1)²] [Using distance formula]
= √[(4 - (-2))² + (6 - 8)²] [Substituting the coordinates of R(4,6) and P(-2,8)]
= √[6² + (-2)²]
= √[36 + 4]
= √40
= 2√10
Now let's analyze the angles:
1. Angle PQR:
We can find this angle by using the slopes of two sides: PQ and QR.
Slope of PQ (m1) = (y2 - y1)/(x2 - x1) = (4 - 8)/(2 - (-2)) = -4/4 = -1
Slope of QR (m2) = (y2 - y1)/(x2 - x1) = (6 - 4)/(4 - 2) = 2/2 = 1
Since the slopes are negative reciprocals of each other, the two sides PQ and QR are perpendicular to each other. Therefore, angle PQR is a right angle (∠PQR = 90°).
2. Angle PQP and Angle QPR:
We can observe that PR is the longest side among the three sides, so it cannot be a right triangle with PQ and PR or PQ and QR as the legs. Additionally, the lengths of sides PQ and QR (4√2 and 2√2 respectively) are not the same. Therefore, triangle PQR is not an isosceles triangle.
Based on the analysis, the best description for triangle PQR is:
A- a right triangle with PQ ㅗ QR
first, find the side lengths.
PQ = √(4^2+4^2) = √32
Find the others, and then you can see which kind of triangle it is.
Think Pythagorean Theorem