Show that the points (1, 1), (5, 3) and (6, -4) are vertices of an isosceles triangle, and find one of the equal angles.
To show that the points (1, 1), (5, 3) and (6, -4) are vertices of an isosceles triangle, we need to show that two sides of the triangle are equal.
Let's use the distance formula to find the lengths of the sides. The distance between (1, 1) and (5, 3) is:
√((5-1)^2 + (3-1)^2) = √(16 + 4) = √20 = 2√5
The distance between (1, 1) and (6, -4) is:
√((6-1)^2 + (-4-1)^2) = √(25 + 25) = √50 = 5√2
The distance between (5, 3) and (6, -4) is:
√((6-5)^2 + (-4-3)^2) = √(1 + 49) = √50 = 5√2
We can see that two sides of the triangle have the same length, which means the triangle is isosceles.
Now let's find one of the equal angles in the triangle. We can use the slope formula to find the slopes of the sides. The slope between (1, 1) and (5, 3) is:
(3-1) / (5-1) = 2/4 = 1/2
The slope between (1, 1) and (6, -4) is:
(-4-1) / (6-1) = -5/5 = -1
The slope between (5, 3) and (6, -4) is:
(-4-3) / (6-5) = -7/1 = -7
Since two sides of the triangle are equal, the corresponding slopes are equal. So, the slopes -1 and -7 form two sides with equal slopes, which means the corresponding angles are equal.
Therefore, the triangle with vertices (1, 1), (5, 3), and (6, -4) is an isosceles triangle, and one of the equal angles is formed by the sides connecting (1, 1) and (6, -4).
To show that the points (1, 1), (5, 3), and (6, -4) are vertices of an isosceles triangle, we need to find the lengths of the sides of the triangle and determine if any two sides have the same length.
Step 1: Find the lengths of the sides using the distance formula.
The distance formula is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Using this formula, we can find the lengths of the three sides of the triangle.
Side 1: Length between (1, 1) and (5, 3)
d1 = sqrt((5 - 1)^2 + (3 - 1)^2)
= sqrt(4^2 + 2^2)
= sqrt(16 + 4)
= sqrt(20)
= 2 * sqrt(5)
Side 2: Length between (5, 3) and (6, -4)
d2 = sqrt((6 - 5)^2 + (-4 - 3)^2)
= sqrt(1^2 + (-7)^2)
= sqrt(1 + 49)
= sqrt(50)
= 5 * sqrt(2)
Side 3: Length between (6, -4) and (1, 1)
d3 = sqrt((1 - 6)^2 + (1 - (-4))^2)
= sqrt((-5)^2 + (1 + 4)^2)
= sqrt(25 + 25)
= sqrt(50)
= 5 * sqrt(2)
Step 2: Determine if any two sides have the same length.
From the calculations above, we can see that side 2 and side 3 have the same length, each being 5 * sqrt(2). Therefore, the points (1, 1), (5, 3), and (6, -4) form an isosceles triangle.
Step 3: Find one of the equal angles.
To find the equal angles in an isosceles triangle, we can use the Law of Cosines.
The Law of Cosines states that in a triangle with sides a, b, and c, and angle C opposite side c, the following formula holds:
c^2 = a^2 + b^2 - 2ab * cos(C)
Let's use this formula to find one of the equal angles.
Using side 1 (Length = 2 * sqrt(5)), side 2 (Length = 5 * sqrt(2)), and side 3 (Length = 5 * sqrt(2)) as a, b, and c respectively, we can apply the Law of Cosines as follows:
(5 * sqrt(2))^2 = (2 * sqrt(5))^2 + (5 * sqrt(2))^2 - 2 * (2 * sqrt(5)) * (5 * sqrt(2)) * cos(C)
50 = 20 + 50 - 20 * cos(C)
50 - 70 = -20 * cos(C)
-20 = -20 * cos(C)
cos(C) = 1
Since cos(C) = 1 implies that C = 0 degrees, one of the equal angles in the triangle is 0 degrees.
Therefore, the points (1, 1), (5, 3), and (6, -4) are vertices of an isosceles triangle, and one of the equal angles is 0 degrees.
To show that the points (1, 1), (5, 3), and (6, -4) are vertices of an isosceles triangle and find one of the equal angles, we need to calculate the lengths of the sides of the triangle. Then, we can compare the lengths to determine if the triangle is isosceles. Finally, we can find the equal angle by using the properties of isosceles triangles.
Step 1: Calculate the lengths of the sides:
- Length of side AB = √[(x2 - x1)^2 + (y2 - y1)^2]
- Length of side BC = √[(x3 - x2)^2 + (y3 - y2)^2]
- Length of side AC = √[(x3 - x1)^2 + (y3 - y1)^2]
Using the given points:
- Length of side AB = √[(5 - 1)^2 + (3 - 1)^2] = √[16 + 4] = √20 = 2√5
- Length of side BC = √[(6 - 5)^2 + (-4 - 3)^2] = √[1 + 49] = √50 = 5√2
- Length of side AC = √[(6 - 1)^2 + (-4 - 1)^2] = √[25 + 25] = √50 = 5√2
Step 2: Compare the lengths of the sides:
We can see that the lengths of sides AB and AC are both equal to 2√5, while the length of side BC is equal to 5√2. Since two sides have the same length, the triangle is isosceles.
Step 3: Find one of the equal angles:
To find one of the equal angles, we can use the fact that in an isosceles triangle, the angles opposite the equal sides are congruent. In our case, angles A and C are equal.
To find the measure of angle A (or angle C), we can use the formula:
- cos(A) = (b^2 + c^2 - a^2) / (2bc), where a, b, and c are the lengths of sides AC, AB, and BC, respectively.
Using the lengths of the sides we calculated earlier:
- cos(A) = (5√2)^2 + (5√2)^2 - (2√5)^2) / (2 * 5√2 * 5√2)
- cos(A) = (50 + 50 - 20) / (2 * 25 * 2)
- cos(A) = 80 / 100
- cos(A) = 0.8
Now, using the inverse cosine function, we can find the measure of angle A:
- A = cos^(-1)(0.8)
- A ≈ 36.87 degrees
Therefore, one of the equal angles in the isosceles triangle with vertices (1, 1), (5, 3), and (6, -4) is approximately 36.87 degrees.