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).
Show that the points (1, 1), (5, 3) and (6, -4) are vertices of an isosceles triangle, and find one of the equal angles.
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