Determine whether ∆QRS is a right triangle for the given vertices. Explain.
Q(18, 13), R(17, –3), S(–18, 12)
There must be a 90° angle, that is, two of the lines must be perpendicular, or
the slopes of two of the lines must be negative reciprocals of each other.
So ....
take the slope of the 3 lines and check for the above property.
A longer method would be to find the lengths of the 3 line segments, and check using Pythagoras.
To determine whether ∆QRS is a right triangle, we need to check if any of the sides of the triangle are perpendicular to each other.
The formula to calculate the slope between two points (x1, y1) and (x2, y2) is:
slope = (y2 - y1) / (x2 - x1)
Let's calculate the slopes of the sides QR, RS, and SQ:
1. Slope of QR:
Q(18, 13), R(17, -3)
slope QR = (-3 - 13) / (17 - 18) = -16 / -1 = 16
2. Slope of RS:
R(17, -3), S(-18, 12)
slope RS = (12 - -3) / (-18 - 17) = 15 / -35 = -3/7
3. Slope of SQ:
S(-18, 12), Q(18, 13)
slope SQ = (13 - 12) / (18 - -18) = 1 / 36
Now, we need to check if any pair of slopes are negative reciprocals of each other (product of slopes equal to -1). If there is such a pair, the triangle is a right triangle.
Checking the slopes:
- Slopes of QR and RS:
Product = 16 * (-3/7) = -48/7 ≠ -1
- Slopes of RS and SQ:
Product = (-3/7) * (1/36) = -1/84 ≠ -1
- Slopes of SQ and QR:
Product = (1/36) * 16 = 4/9 ≠ -1
Since none of the products of the slopes equal -1, we can conclude that ∆QRS is not a right triangle.