An isosceles traingle has a base of length 2m sides and length of square root 2. Determine the exact value of the apex angle in radius.
To determine the apex angle of the isosceles triangle, we can use the properties of triangles. In an isosceles triangle, the base angles are equal. Let's call the base angles x, and the apex angle y.
We know that the sum of angles in a triangle is 180 degrees, or π radians. Therefore, we can write an equation:
x + x + y = π
Since the base angles are equal, we have:
2x + y = π
Now, we have another property of isosceles triangles - the base angles are congruent to the opposite sides (or vice versa). This means that we can use the lengths of the sides to find the angles.
In our case, the sides of the triangle have lengths of 2 and √2. To find the angles, we can use the inverse trigonometric functions.
Let's consider the base angles x. We can use the inverse tangent function (tan⁻¹) to find x:
tan⁻¹(2/√2) = x
Simplifying the expression:
2/√2 = √2
tan⁻¹(√2) = x
Similarly, we can find the apex angle y using the sides of the triangle:
tan⁻¹(2/√2) + tan⁻¹(√2) + y = π
Simplifying the expression:
π/4 + π/4 + y = π
π/2 + y = π
y = π/2
Therefore, the exact value of the apex angle in radians is π/2.
Sketch your triangle, draw in an altitude to see 2 congruent right-angled triangle
Look at one them.
It has a base of 1, a height of h and a hypotenuse of √2
So you recognize the triangle? You should know this one.
So what is the vertex angle?