The most popular triangle used in construction, engineering and mathematics is the right triangle, which has a 90o angle at the base. We'll call your base "x", height "y" and diagonal (hypotenuse) "r".
You can find these values using your calculator's trigonometric functions, or you can use the Pythagorean Theorem.
Finding Values (Pythagorean Theorem):
The Pythagorean Theorem is straightforward and states that, for a right triangle, r^2 = x^2 + y^2. We can use a little bit of algebra to find x, y and r.
r = sqrt(x^2 + y^2)
r^2 = x^2 + y^2
r = sqrt(x^2 + y^2); remove the ^2 from r
x = sqrt(r^2 - y^2)
r^2 = x^2 + y^2
r^2 - y^2 = x^2; move y^2 to the left side of the equation
sqrt(r^2 - y^2) = x; remove the ^2 from x
y = sqrt(r^2 - x^2)
r^2 = x^2 + y^2
r^2 - x^2 = y^2; move x^2 to the left side of the equation
sqrt(r^2 - x^2) = y; remove the ^2 from y
Finding Values (Trigonometric Functions):
Acquiring Ratios:
cos(angle) = x/r
sin(angle) = y/r
tan(angle) = y/x
Acquiring Angles:
cos-1(x/r) = angle
sin-1(y/r) = angle
tan-1(y/x) = angle
That looks confusing. What are cos, sin, tan, etc?
They're functions. You give a function input and it outputs something else. How they do this is complicated and required calculus.
x = cos(angle) * r
cos(angle) = x/r
cos(angle) * r = x; move r to the left side of the equation
y = sin(angle) * r
sin(angle) = y/r
sin(angle) * r = y; move r to the left side of the equation
x = 1 / (tan(angle)/y)
tan(angle) = y/x
tan(angle) / y = 1/x; move y to the left side of the equation
1 / (tan(angle) / y) = x; flip the equation
y = tan(angle) * x
tan(angle) = y/x
tan(angle) * x = y; move x to the left side of the equation
r = 1 / (cos(angle)/x)
cos(angle) = x/r
cos(angle) / x = 1/r; move x to the left side of the equation
1 / (cos(angle)/x) = r; flip the equation
r = 1 / (sin(angle)/y)
sin(angle) = y/r
sin(angle) / y = 1/r; move y to the left side of the equation
1 / (sin(angle)/y) = r; flip the equation
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if the equal sides of an isosceles triangle have lengths a, and the third side is 12 in., find a
-- Find the length of one side. -- Find the length of another side. -- Find the length of the remaining side. -- Add the three numbers. -- Their sum is the perimeter of the scalene triangle.
If the triangle is equilateral, you simply divide the perimeter by three to find the length of each side. If the triangle is not equilateral, you will need more information to determine the length of each side.
To find the altitude or height of an equilateral triangle, take one-half of the length of a side of the triangle and multiple by "square root" of 3. So, if for example, the side has length 10, the height = 5 Square root of 3.
To find side lengths on a triangle, you need to know at least one of the sides. The possible combinations for solving* a triangle are: side, side, side; side, angle, side; angle, side, angle; angle, side, longer side. *To solve a triangle is to find the lengths of all the sides and the measures of all the angles.
ABC is an equilateral triangle with side length equal to 50 cm. BH is perpendicular to AC. MN is parallel to AC. Find the area of triangle BMN if the length of MN is equal to 12 cm.