Determine whether QRS is a right triangle for the given vertices. Explain.
Q(12, 15), R(18, 15), S(12, 7)
A. yes; QR = , QS = , RS = ; RS^2 + QS^2 = RQ^2
B. no; QR = , QS = , RS = ; QR^2 + QS^2 ≠ RS^2
C. yes; QR = , QS = , RS = ; QR^2 + QS^2 = RS^2*
D. no; QR = , QS = , RS = ; RS^2 + QS^2 ≠ RQ^2
On QR, the x co-ordinate increases from 12 to 18 while the y co-ordinate
remains constant at 15. Therefore, QR is a horizontal line.
On QS, the x co-ordinate remains constant. therefore, QS is a vertical line.
So QR and QS are perpendicular and form a rt. angle. Therefore, we have
a rt. triangle with hyp. RS.
To determine whether QRS is a right triangle, we can use the distance formula and the Pythagorean theorem.
First, let's find the lengths of each side of the triangle.
QR = √((18 - 12)^2 + (15 - 15)^2) = √(6^2 + 0^2) = √(36 + 0) = √36 = 6
QS = √((12 - 12)^2 + (7 - 15)^2) = √(0^2 + (-8)^2) = √(0 + 64) = √64 = 8
RS = √((12 - 18)^2 + (7 - 15)^2) = √((-6)^2 + (-8)^2) = √(36 + 64) = √100 = 10
Now, let's square each side and see if the Pythagorean theorem holds:
QR^2 + QS^2 = 6^2 + 8^2 = 36 + 64 = 100
RS^2 = 10^2 = 100
Since QR^2 + QS^2 = RS^2, the Pythagorean theorem holds, which means that QRS is a right triangle. Therefore, the correct answer is:
C. yes; QR = 6, QS = 8, RS = 10; QR^2 + QS^2 = RS^2
To determine whether QRS is a right triangle, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
In this case, the given vertices are Q(12, 15), R(18, 15), and S(12, 7). To find the lengths of the sides, we can use the distance formula.
The distance formula between two points (x1, y1) and (x2, y2) is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Using the distance formula, let's calculate the lengths of the sides:
1. QR:
QR = √((18 - 12)^2 + (15 - 15)^2)
= √(6^2 + 0^2)
= √(36 + 0)
= √36
= 6
2. QS:
QS = √((12 - 12)^2 + (7 - 15)^2)
= √(0^2 + (-8)^2)
= √(0 + 64)
= √64
= 8
3. RS:
RS = √((18 - 12)^2 + (15 - 7)^2)
= √(6^2 + 8^2)
= √(36 + 64)
= √100
= 10
Now, we can check if QRS is a right triangle by comparing the squares of the lengths of the sides:
RS^2 + QS^2 = QR^2
10^2 + 8^2 = 6^2
100 + 64 = 36
164 ≠ 36
Since RS^2 + QS^2 is not equal to QR^2, we can conclude that QRS is not a right triangle. Therefore, the correct answer is (B) no; QR^2 + QS^2 ≠ RS^2.