list the sides in order from shortest to longest in angle PQR,with P=45,Q=10x+30,and R=5x
To list the sides in order from shortest to longest in triangle PQR, where P = 45°, Q = 10x + 30, and R = 5x, we need to determine the values of x, substitute them into the expressions for sides PQ and PR, and then compare the lengths.
Let's start by using the triangle angle sum property, which states that the sum of all angles in a triangle is always 180°. Therefore, we can write the equation:
P + Q + R = 180
Substituting the given values for P and R:
45 + (10x + 30) + 5x = 180
Combining like terms:
15x + 75 = 180
Now, let's solve this equation for x:
15x = 180 - 75
15x = 105
x = 7
Now that we have found the value of x, we can substitute it back into the expressions for sides PQ and PR:
PQ = 10x + 30
PQ = 10(7) + 30
PQ = 70 + 30
PQ = 100
PR = 5x
PR = 5(7)
PR = 35
Now, we can compare the lengths of sides PQ and PR. Since PQ = 100 and PR = 35, we can conclude that side PR is the shortest, and side PQ is the longest.
Therefore, listing the sides in order from shortest to longest in triangle PQR, we have:
PR < QR < PQ
(Note: Since the value of x was found to be 7, you may further verify the values of the sides using these values.)