in triangle pqr,if st=qr then find x
Answers
How to find the sides of a right triangle
There are a few methods of obtaining right triangle side lengths. Depending on what is given, you can use different relationships or laws to find the missing side:
Given two sides
If you know two other sides of the right triangle, it's the easiest option; all you need to do is apply the Pythagorean theorem:
a² + b² = c²
if leg a is the missing side, then transform the equation to the form when a is on one side, and take a square root:
a = √(c² - b²)
if leg b is unknown, then
b = √(c² - a²)
for hypotenuse c missing, the formula is
c = √(a² + b²)
Given angle and hypotenuse
Right triangle with law of sines formulas. a over sin(α) equals b over sin(β) equals c, because sin(90°) = 1
Apply the law of sines or trigonometry to find the right triangle side lengths:
a = c * sin(α) or a = c * cos(β)
b = c * sin(β) or b = c * cos(α)
Given angle and one leg
Find the missing leg using trigonometric functions:
a = b * tan(α)
b = a * tan(β)
Given area and one leg
As we remember from basic triangle area formula, we can calculate the area by multiplying triangle height and base and dividing the result by two. A right triangle is a special case of a scalene triangle, in which one leg is the height when the second leg is the base, so the equation gets simplified to:
area = a * b / 2
For example, if we know only the right triangle area and the length of the leg a, we can derive the equation for other sides:
b = 2 * area / a
c = √(a² + (2 * area / a)²)
How to find the angle of a right triangle
If you know one angle apart from the right angle, calculation of the third one is a piece of cake:
Givenβ: α = 90 - β
Givenα: β = 90 - α
However, if only two sides of a triangle are given, finding the angles of a right triangle requires applying some basic trigonometric functions:
for α
sin(α) = a / c so α = arcsin(a / c) (inverse sine)
cos(α) = b / c so α = arccos(b / c) (inverse cosine)
tan(α) = a / b so α = arctan(a / b) (inverse tangent)
cot(α) = b / a so α = arccot(b / a) (inverse cotangent)
and for β
sin(β) = b / c so β = arcsin(b / c) (inverse sine)
cos(β) = a / c so β = arccos(a / c) (inverse cosine)
tan(β) = b / a so β = arctan(b / a) (inverse tangent)
cot(β) = a / b so β = arccot(a / b) (inverse cotangent)