Question
The coordinates of the vertices of the triangle shown are P(−7, 6), Q(5, −4), and R(−7, −4). Which measurement is closest to the length of PQ in units?
The image shows a right-angled triangle plotted on a Cartesian coordinate plane. The vertices of the triangle are labeled P, Q, and R. The coordinates of these points are:
P (-7, 6)
Q (3, -3)
R (-7, -3)
The triangle's sides are aligned with the grid lines, making it easy to determine their lengths. The horizontal side RQ has a length of 10 units (from -7 to 3 on the x-axis), and the vertical side RP has a length of 9 units (from -3 to 6 on the y-axis). The hypotenuse PQ is the longest side, connecting points P and Q diagonally across the grid. The coordinate plane itself ranges from -10 to 10 on both the x and y axes, with grid lines at every unit.
Responses
A 14.4 units14.4 units
B 14.8 units14.8 units
C 15.2 units15.2 units
D 15.6 units
The image shows a right-angled triangle plotted on a Cartesian coordinate plane. The vertices of the triangle are labeled P, Q, and R. The coordinates of these points are:
P (-7, 6)
Q (3, -3)
R (-7, -3)
The triangle's sides are aligned with the grid lines, making it easy to determine their lengths. The horizontal side RQ has a length of 10 units (from -7 to 3 on the x-axis), and the vertical side RP has a length of 9 units (from -3 to 6 on the y-axis). The hypotenuse PQ is the longest side, connecting points P and Q diagonally across the grid. The coordinate plane itself ranges from -10 to 10 on both the x and y axes, with grid lines at every unit.
Responses
A 14.4 units14.4 units
B 14.8 units14.8 units
C 15.2 units15.2 units
D 15.6 units
Answers
GPT-4o mini
To find the length of \( PQ \) between the points \( P(-7, 6) \) and \( Q(5, -4) \), we can use the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Where \( (x_1, y_1) \) are the coordinates of point \( P \) and \( (x_2, y_2) \) are the coordinates of point \( Q \).
Substituting the coordinates \( P(-7, 6) \) and \( Q(5, -4) \) into the formula, we have:
- \( x_1 = -7 \)
- \( y_1 = 6 \)
- \( x_2 = 5 \)
- \( y_2 = -4 \)
Now, calculate the differences:
\[
x_2 - x_1 = 5 - (-7) = 5 + 7 = 12
\]
\[
y_2 - y_1 = -4 - 6 = -10
\]
Now plug these values into the distance formula:
\[
d = \sqrt{(12)^2 + (-10)^2}
\]
Calculating the squares:
\[
d = \sqrt{144 + 100} = \sqrt{244}
\]
Now, we can simplify \( \sqrt{244} \):
\[
\sqrt{244} = \sqrt{4 \cdot 61} = 2\sqrt{61}
\]
Next, we can approximate \( \sqrt{61} \):
\[
\sqrt{61} \approx 7.81 \quad (\text{since } 7.81^2 \approx 61)
\]
Hence, we find:
\[
d \approx 2 \cdot 7.81 = 15.62
\]
Comparing this result to the given options:
- A: 14.4 units
- B: 14.8 units
- C: 15.2 units
- D: 15.6 units
The closest measurement to \( 15.62 \) is \( D: 15.6 \) units.
Thus, the correct answer is:
**D: 15.6 units**
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Where \( (x_1, y_1) \) are the coordinates of point \( P \) and \( (x_2, y_2) \) are the coordinates of point \( Q \).
Substituting the coordinates \( P(-7, 6) \) and \( Q(5, -4) \) into the formula, we have:
- \( x_1 = -7 \)
- \( y_1 = 6 \)
- \( x_2 = 5 \)
- \( y_2 = -4 \)
Now, calculate the differences:
\[
x_2 - x_1 = 5 - (-7) = 5 + 7 = 12
\]
\[
y_2 - y_1 = -4 - 6 = -10
\]
Now plug these values into the distance formula:
\[
d = \sqrt{(12)^2 + (-10)^2}
\]
Calculating the squares:
\[
d = \sqrt{144 + 100} = \sqrt{244}
\]
Now, we can simplify \( \sqrt{244} \):
\[
\sqrt{244} = \sqrt{4 \cdot 61} = 2\sqrt{61}
\]
Next, we can approximate \( \sqrt{61} \):
\[
\sqrt{61} \approx 7.81 \quad (\text{since } 7.81^2 \approx 61)
\]
Hence, we find:
\[
d \approx 2 \cdot 7.81 = 15.62
\]
Comparing this result to the given options:
- A: 14.4 units
- B: 14.8 units
- C: 15.2 units
- D: 15.6 units
The closest measurement to \( 15.62 \) is \( D: 15.6 \) units.
Thus, the correct answer is:
**D: 15.6 units**