The triangle DEF has vertices D(-1,3)
E(7,1) and F(4,6). Classify the triangle as:
a) isosceles or scalene
b) right angled or not
L=√(8²+2²)=√68
To classify the triangle DEF, we first need to determine its side lengths and angle measurements.
a) To determine if the triangle is isosceles or scalene, we need to find the lengths of the three sides.
Using the distance formula, the lengths can be calculated as follows:
Length DE:
√[(7 - (-1))^2 + (1 - 3)^2] = √[8^2 + (-2)^2] = √[64 + 4] = √68 ≈ 8.246
Length EF:
√[(4 - 7)^2 + (6 - 1)^2] = √[(-3)^2 + 5^2] = √[9 + 25] = √34 ≈ 5.831
Length FD:
√[(-1 - 4)^2 + (3 - 6)^2] = √[(-5)^2 + (-3)^2] = √[25 + 9] = √34 ≈ 5.831
Comparing the lengths, DE is approximately 8.246, and both EF and FD are approximately 5.831. Since all three sides have different lengths, the triangle DEF is classified as scalene.
b) To determine if the triangle is right-angled or not, we can examine the angles. One way to find the angle measurements is by using the dot product formula:
Let vector AB be the difference between points A and B:
AB = (x2 - x1, y2 - y1)
The dot product of two vectors A and B is given by:
A · B = (x1 * x2) + (y1 * y2)
If the dot product of two vectors is zero, it means the vectors are orthogonal, which implies that the corresponding sides are perpendicular.
Let's calculate the dot products for the sides of the triangle:
DE · EF = (-8 * -3) + (-2 * 5) = 24 - 10 = 14
EF · FD = (-3 * 5) + (5 * -3) = -15 - 15 = -30
FD · DE = (-5 * -8) + (-3 * -2) = 40 - 6 = 34
None of the dot products are zero, which means none of the sides are perpendicular and the triangle DEF is not right-angled.
Therefore, the triangle DEF is classified as:
a) scalene
b) not right-angled.
To classify the triangle DEF as either isosceles or scalene, and as either right-angled or not, we need to examine its sides and angles.
a) Isosceles or Scalene:
To determine if the triangle is isosceles or scalene, we need to check the lengths of its sides.
Step 1: Calculate the lengths of the three sides of the triangle:
- Side DE:
Using the distance formula, we can find the distance between points D(-1, 3) and E(7, 1):
DE = √[(x2 - x1)^2 + (y2 - y1)^2]
DE = √[(7 - (-1))^2 + (1 - 3)^2]
DE = √[8^2 + (-2)^2]
DE = √[64 + 4]
DE = √68
DE ≈ 8.246
- Side EF:
Using the distance formula, we can find the distance between points E(7, 1) and F(4, 6):
EF = √[(x2 - x1)^2 + (y2 - y1)^2]
EF = √[(4 - 7)^2 + (6 - 1)^2]
EF = √[(-3)^2 + 5^2]
EF = √[9 + 25]
EF = √34
EF ≈ 5.831
- Side FD:
Using the distance formula, we can find the distance between points F(4, 6) and D(-1, 3):
FD = √[(x2 - x1)^2 + (y2 - y1)^2]
FD = √[(-1 - 4)^2 + (3 - 6)^2]
FD = √[(-5)^2 + (-3)^2]
FD = √[25 + 9]
FD = √34
FD ≈ 5.831
Step 2: Compare the lengths of the sides:
Since DE ≈ 8.246, EF ≈ 5.831, and FD ≈ 5.831, we can conclude that no two sides of the triangle are of the same length. Therefore, the triangle DEF is classified as a scalene triangle.
b) Right-angled or not:
To determine if the triangle is right-angled or not, we need to check its angles.
Step 1: Calculate the slopes of the two sides:
- Side DE:
The formula for the slope (m) between two points (x1, y1) and (x2, y2) is given by:
m = (y2 - y1) / (x2 - x1)
m(DE) = (1 - 3) / (7 - (-1))
m(DE) = -2 / 8
m(DE) = -1/4
- Side EF:
m(EF) = (6 - 1) / (4 - 7)
m(EF) = 5 / (-3)
m(EF) = -5/3
Step 2: Check if the slopes are negative reciprocals of each other:
Since m(DE) and m(EF) are not negative reciprocals of each other (-1/4 and -5/3 are not reciprocals), the triangle DEF is not a right-angled triangle.
In conclusion, the triangle DEF is classified as a scalene triangle and is not right-angled.
An isosceles triangle has at least two equal sides.
To check this, we find the difference between the end-points of each of the sides, and from that, we calculate the length using Pythagoras Theorem.
D-E (-1,3)-(7,1)=(-1-7,3-1)=(-8,2)
L=√(8²+2*sup2;)=√68
E-F (7,1)-(4,6)=(7-4,1-6)=(3,-5)
L=√(3²+(-5)²)=√34
F-D (4,6)-(-1,3)=(4-(-1),6-3)=(5,3)
L=√(5²+3²)=√34
Since mEF=mFD, we conclude that the triangle DEF is isosceles.
Since mEF²+mFD²=mDE², we conclude that ∠EFD is a right angle, thus the triangle is a right-triangle.