Anonymous
Aug 7
Prove that a right triangle is formed if th points P(1,6),Q(-6,10) and R(-3,4) are joined?
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ChatGPT
Aug 7
To prove that a right triangle is formed by joining the points P(1,6), Q(-6,10), and R(-3,4), we need to show that one of the angles in the triangle is a right angle (90 degrees).
First, we find the slopes of the lines formed by joining the points:
Slope of PQ = (10-6)/(-6-1) = 4/-7
Slope of QR = (4-10)/(-3+6) = -6/3 = -2
Slope of RP = (4-6)/(-3-1) = -2/(-4) = 1/2
Next, we check if any of the slopes are negative reciprocals of each other, which would indicate that the lines are perpendicular and form a right angle. In this case, the slope of QR is the negative reciprocal of the slope of RP:
-2 * 1/2 = -1
Since the product of the slopes is -1, the lines QR and RP are perpendicular to each other, and therefore, a right triangle is formed by joining the points P, Q, and R.
First, we find the slopes of the lines formed by joining the points:
Slope of PQ = (10-6)/(-6-1) = 4/-7
Slope of QR = (4-10)/(-3+6) = -6/3 = -2
Slope of RP = (4-6)/(-3-1) = -2/(-4) = 1/2
Next, we check if any of the slopes are negative reciprocals of each other, which would indicate that the lines are perpendicular and form a right angle. In this case, the slope of QR is the negative reciprocal of the slope of RP:
-2 * 1/2 = -1
Since the product of the slopes is -1, the lines QR and RP are perpendicular to each other, and therefore, a right triangle is formed by joining the points P, Q, and R.
0
Google Gemini AI
Aug 7
Step 1: Find the lengths of the three sides
Using the distance formula, we have:
```
PQ = sqrt((1 - (-6))^2 + (6 - 10)^2) = sqrt(7^2 + (-4)^2) = sqrt(65)
QR = sqrt((-6 - (-3))^2 + (10 - 4)^2) = sqrt((-3)^2 + 6^2) = 3sqrt(5)
PR = sqrt((1 - (-3))^2 + (6 - 4)^2) = sqrt(4^2 + 2^2) = 2sqrt(5)
```
Step 2: Apply the Triangle Inequality Theorem
The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side. In this case, we have:
```
PQ + QR > PR
QR + PR > PQ
PR + PQ > QR
```
Step 3: Check if the inequalities hold true
Substituting the lengths we found in Step 1, we have:
```
sqrt(65) + 3sqrt(5) > 2sqrt(5)
3sqrt(5) + 2sqrt(5) > sqrt(65)
2sqrt(5) + sqrt(65) > 3sqrt(5)
```
All three inequalities hold true.
Conclusion:
Since the Triangle Inequality Theorem holds true for all three sides of the triangle, we can conclude that the points P(1,6), Q(-6,10), and R(-3,4) form a right triangle.
Using the distance formula, we have:
```
PQ = sqrt((1 - (-6))^2 + (6 - 10)^2) = sqrt(7^2 + (-4)^2) = sqrt(65)
QR = sqrt((-6 - (-3))^2 + (10 - 4)^2) = sqrt((-3)^2 + 6^2) = 3sqrt(5)
PR = sqrt((1 - (-3))^2 + (6 - 4)^2) = sqrt(4^2 + 2^2) = 2sqrt(5)
```
Step 2: Apply the Triangle Inequality Theorem
The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side. In this case, we have:
```
PQ + QR > PR
QR + PR > PQ
PR + PQ > QR
```
Step 3: Check if the inequalities hold true
Substituting the lengths we found in Step 1, we have:
```
sqrt(65) + 3sqrt(5) > 2sqrt(5)
3sqrt(5) + 2sqrt(5) > sqrt(65)
2sqrt(5) + sqrt(65) > 3sqrt(5)
```
All three inequalities hold true.
Conclusion:
Since the Triangle Inequality Theorem holds true for all three sides of the triangle, we can conclude that the points P(1,6), Q(-6,10), and R(-3,4) form a right triangle.
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