Question

determine whether a triangle with the given vertices is a scalene, isosceles, or equilateral triangle. check all that apply.

	scalene triangle	isosceles triangle	equilateral triangle
(b) ( d(-2, 3) ), ( e(-6, 4) ), ( f(0, 4) )	( square )	( square )	( square )
(c) ( p(1, -3) ), ( q(0, 2) ), ( r(-4, -3) )	( square )	( square )	( square )

Explanation:

To determine the type of triangle, we calculate the lengths of the sides using the distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).

Part (a): Vertices \( A(0, -3) \), \( B(5, 1) \), \( C(5, -7) \)

Step 1: Calculate \( AB \)

\( AB = \sqrt{(5 - 0)^2 + (1 - (-3))^2} = \sqrt{25 + 16} = \sqrt{41} \)

Step 2: Calculate \( BC \)

\( BC = \sqrt{(5 - 5)^2 + (-7 - 1)^2} = \sqrt{0 + 64} = 8 \)

Step 3: Calculate \( AC \)

\( AC = \sqrt{(5 - 0)^2 + (-7 - (-3))^2} = \sqrt{25 + 16} = \sqrt{41} \)
Since \( AB = AC = \sqrt{41} \) and \( BC = 8 \), two sides are equal. So it is an isosceles triangle. It is not scalene (all sides different) or equilateral (all sides equal).

Part (b): Vertices \( D(-2, 3) \), \( E(-6, 4) \), \( F(0, 4) \)

Step 1: Calculate \( DE \)

\( DE = \sqrt{(-6 - (-2))^2 + (4 - 3)^2} = \sqrt{16 + 1} = \sqrt{17} \)

Step 2: Calculate \( EF \)

\( EF = \sqrt{(0 - (-6))^2 + (4 - 4)^2} = \sqrt{36 + 0} = 6 \)

Step 3: Calculate \( DF \)

\( DF = \sqrt{(0 - (-2))^2 + (4 - 3)^2} = \sqrt{4 + 1} = \sqrt{5} \)
All sides \( \sqrt{17} \), \( 6 \), \( \sqrt{5} \) are different. So it is a scalene triangle. It is not isosceles or equilateral.

Part (c): Vertices \( P(1, -3) \), \( Q(0, 2) \), \( R(-4, -3) \)

Step 1: Calculate \( PQ \)

\( PQ = \sqrt{(0 - 1)^2 + (2 - (-3))^2} = \sqrt{1 + 25} = \sqrt{26} \)

Step 2: Calculate \( QR \)

\( QR = \sqrt{(-4 - 0)^2 + (-3 - 2)^2} = \sqrt{16 + 25} = \sqrt{41} \)

Step 3: Calculate \( PR \)

\( PR = \sqrt{(-4 - 1)^2 + (-3 - (-3))^2} = \sqrt{25 + 0} = 5 \)
All sides \( \sqrt{26} \), \( \sqrt{41} \), \( 5 \) are different. Wait, no—wait, \( P(1, -3) \) and \( R(-4, -3) \): the y-coordinates are the same, so \( PR = |-4 - 1| = 5 \). Wait, recalculating \( PQ \): \( (0 - 1)^2 + (2 - (-3))^2 = 1 + 25 = 26 \), so \( PQ = \sqrt{26} \approx 5.1 \). \( QR \): \( (-4 - 0)^2 + (-3 - 2)^2 = 16 + 25 = 41 \), so \( QR = \sqrt{41} \approx 6.4 \). \( PR = 5 \). Wait, but also, check \( P(1, -3) \) and \( R(-4, -3) \), \( Q(0, 2) \): let's check \( PQ \) and \( RQ \)? Wait no, original vertices: \( P(1, -3) \), \( Q(0, 2) \), \( R(-4, -3) \). Wait, \( P \) and \( R \) have the same y-coordinate, so \( PR = 5 \). \( PQ \): distance from \( (1, -3) \) to \( (0, 2) \): \( \sqrt{(0 - 1)^2 + (2 - (-3))^2} = \sqrt{1 + 25} = \sqrt{26} \). \( RQ \): distance from \( (-4, -3) \) to \( (0, 2) \): \( \sqrt{(0 - (-4))^2 + (2 - (-3))^2} = \sqrt{16 + 25} = \sqrt{41} \). Wait, but also, check \( P(1, -3) \) and \( R(-4, -3) \), \( Q(0, 2) \): is there a mistake? Wait, no—wait, \( P(1, -3) \) and \( R(-4, -3) \): the distance is \( |1 - (-4)| = 5 \) (since y is same). \( P(1, -3) \) to \( Q(0, 2) \): \( \sqrt{1 + 25} = \sqrt{26} \). \( R(-4, -3) \) to \( Q(0, 2) \): \( \sqrt{16 + 25} = \sqrt{41} \). Wait, but also, check \( PQ \) and \( RQ \)? No, the sides are \( PQ \), \( QR \), \( PR \). Wait, but wait—\( P(1, -3) \) and \( R(-4, -3) \): y-coordinates are equal, so horizontal line. \( Q(0, 2) \): let's see the distance from \( Q \) to \( P \) and \( Q \) to \( R \). Wait, \( PQ = \sqrt{26} \), \( RQ = \sqrt{41} \), \( PR = 5 \). Wait, but also, check \( P(1, -3) \) and \( R(-4, -3) \): the length is 5. \( P(1, -3) \) to \( Q(0, 2) \): \( \sqrt{26} \approx 5.1 \), which is almost 5, but not equal. Wait, no—wait, maybe I made a mistake. Wait, \( P(1, -3) \), \( R(-4, -3) \): the x-coordinates are 1 and -4, so difference is 5, y same, so length 5. \( P(1, -3) \) to \( Q(0, 2) \): \( \sqrt{(0 - 1)^2 + (2 - (-3))^2} = \sqrt{1 + 25} = \sqrt{26} \approx 5.099 \), which is approximately 5.1, not 5. \( R(-4, -3) \) to \( Q(0, 2) \): \( \sqrt{(0 + 4)^2 + (2 + 3)^2} = \sqrt{16 + 25} = \sqrt{41} \approx 6.403 \). Wait, but maybe I miscalculated \( PR \). Wait, \( P(1, -3) \), \( R(-4, -3) \): \( x_2 - x_1 = -4 - 1 = -5 \), \( y_2 - y_1 = -3 - (-3) = 0 \), so distance is \( \sqrt{(-5)^2 + 0^2} = 5 \). Correct. Then \( PQ = \sqrt{26} \approx 5.099 \), which is not equal to 5, but very close. Wait, but maybe it's a typo? No, the coordinates are given. Wait, but maybe I made a mistake in the sides. Wait, the triangle has vertices \( P(1, -3) \), \( Q(0, 2) \), \( R(-4, -3) \). So sides: \( PQ \), \( QR \), \( PR \). \( PR = 5 \), \( PQ = \sqrt{26} \approx 5.1 \), \( QR = \sqrt{41} \approx 6.4 \). Wait, but also, check \( P(1, -3) \) and \( R(-4, -3) \), \( Q(0, 2) \): is \( PQ = RQ \)? No, \( PQ = \sqrt{26} \), \( RQ = \sqrt{41} \). Wait, but wait—\( P(1, -3) \) and \( R(-4, -3) \): the midpoint is \( (-1.5, -3) \), and \( Q(0, 2) \): is the distance from \( Q \) to \( P \) and \( Q \) to \( R \) equal? No. Wait, maybe I made a mistake. Wait, let's recalculate \( PR \): \( P(1, -3) \), \( R(-4, -3) \): \( \sqrt{( -4 - 1)^2 + (-3 - (-3))^2} = \sqrt{(-5)^2 + 0} = 5 \). Correct. \( PQ \): \( \sqrt{(0 - 1)^2 + (2 - (-3))^2} = \sqrt{1 + 25} = \sqrt{26} \approx 5.099 \). \( QR \): \( \sqrt{(0 - (-4))^2 + (2 - (-3))^2} = \sqrt{16 + 25} = \sqrt{41} \approx 6.403 \). Wait, but maybe the problem is that \( P \) and \( R \) have the same y-coordinate, so \( PR \) is horizontal, and \( Q \) is above. Wait, but the lengths: \( PR = 5 \), \( PQ \approx 5.1 \), \( QR \approx 6.4 \). So all sides are different? But that can't be. Wait, no—wait, \( P(1, -3) \), \( R(-4, -3) \): the distance is 5. \( P(1, -3) \) to \( Q(0, 2) \): \( \sqrt{1 + 25} = \sqrt{26} \approx 5.099 \), which is approximately 5, but not exactly. Wait, maybe the coordinates are \( P(1, -3) \), \( Q(0, 2) \), \( R(-4, -3) \). Wait, \( P \) and \( R \) are 5 units apart. \( P \) to \( Q \): \( \sqrt{26} \approx 5.1 \), \( R \) to \( Q \): \( \sqrt{41} \approx 6.4 \). So all sides are different? But that would be scalene. Wait, but maybe I made a mistake. Wait, let's check \( PQ \) and \( PR \): \( PQ \approx 5.1 \), \( PR = 5 \), not equal. \( QR \approx 6.4 \), not equal to either. So scalene? Wait, but maybe the problem is that \( P(1, -3) \), \( R(-4, -3) \), so \( PR = 5 \), and \( Q(0, 2) \): let's check the distance from \( Q \) to \( P \) and \( Q \) to \( R \) again. \( Q(0, 2) \) to \( P(1, -3) \): \( \sqrt{(1 - 0)^2 + (-3 - 2)^2} = \sqrt{1 + 25} = \sqrt{26} \). \( Q(0, 2) \) to \( R(-4, -3) \): \( \sqrt{(-4 - 0)^2 + (-3 - 2)^2} = \sqrt{16 + 25} = \sqrt{41} \). So all three sides are different: \( 5 \), \( \sqrt{26} \), \( \sqrt{41} \). So scalene. Wait, but maybe I messed up the sides. No, the sides are \( PQ \), \( QR \), \( PR \). So \( PR = 5 \), \( PQ = \sqrt{26} \), \( QR = \sqrt{41} \). All different. So scalene.

Wait, but let's recheck part (c) again. Wait, \( P(1, -3) \), \( R(-4, -3) \): the y-coordinates are equal, so \( PR \) is horizontal, length 5. \( Q(0, 2) \): the distance from \( Q \) to \( P \) is \( \sqrt{(1 - 0)^2 + (-3 - 2)^2} = \sqrt{1 + 25} = \sqrt{26} \). Distance from \( Q \) to \( R \) is \( \sqrt{(-4 - 0)^2 + (-3 - 2)^2} = \sqrt{16 + 25} = \sqrt{41} \). So all three sides are different: \( 5 \), \( \sqrt{26} \), \( \sqrt{41} \). So scalene.

But wait, maybe I made a mistake in part (c). Let's check again. \( P(1, -3) \), \( R(-4, -3) \): \( x \)-difference is \( -4 - 1 = -5 \), \( y \)-difference is \( 0 \), so distance is \( 5 \). Correct. \( P(1, -3) \) to \( Q(0, 2) \): \( x \)-difference \( -1 \), \( y \)-difference \( 5 \), so distance \( \sqrt{1 + 25} = \sqrt{26} \). Correct. \( R(-4, -3) \) to \( Q(0, 2) \): \( x \)-difference \( 4 \), \( y \)-difference \( 5 \), so distance \( \sqrt{16 + 25} = \sqrt{41} \). Correct. So all sides are different: scalene.

Part (b) Recap:

\( DE = \sqrt{17} \), \( EF = 6 \), \( DF = \sqrt{5} \). All different: scalene.

Part (a) Recap:

\( AB = \sqrt{41} \), \( BC = 8 \), \( AC = \sqrt{41} \). Two sides equal: isosceles.

Answers:

(a)

- Scalene triangle: ❌
- Isosceles triangle: ✅
- Equilateral triangle: ❌

(b)

- Scalene triangle: ✅
- Isosceles triangle: ❌
- Equilateral triangle: ❌

(c)

- Scalene triangle: ✅
- Isosceles triangle: ❌
- Equilateral triangle: ❌

(Note: For part (c), if there was a miscalculation, but based on the given coordinates, the sides are all different. However, if we re-express \( \sqrt{26} \approx 5.099 \), \( 5 \), and \( \sqrt{41} \approx 6.403 \), they are all distinct, so scalene.)

Answer:

s:

(a)

• Scalene triangle: ❌
• Isosceles triangle: ✅
• Equilateral triangle: ❌

(b)

• Scalene triangle: ✅
• Isosceles triangle: ❌
• Equilateral triangle: ❌

(c)

• Scalene triangle: ✅
• Isosceles triangle: ❌
• Equilateral triangle: ❌

(Note: For part (c), if there was a miscalculation, but based on the given coordinates, the sides are all different. However, if we re-express \( \sqrt{26} \approx 5.099 \), \( 5 \), and \( \sqrt{41} \approx 6.403 \), they are all distinct, so scalene.)