If all the vertices of a triangle are on the circle and one side of the triangle is the diameter of the circle it is a right angle. Find the length of the diameter (AC) if Ab =8 and BC = 15.
The diameter is the longest side, and the longest side of a rt. triangle is
the hyp.
(AC)^2 = 8^2 + (15)^2 = 289,
AC = 17.
A BIKE RENTAL SHOP CHARGES $8 TO RENT A BIKE, PLUS $1.50 FOE EVERY HALF HOUR YOU RIDE. IF THE SHOP CHARGES YOU AND YOUR FRIEND A TOTAL OF $25, HOW MANY HOURS DID YOU RIDE?
To determine the length of the diameter (AC), let's first understand the given information.
According to the problem, all three vertices of the triangle lie on the circle, and one side of the triangle, BC, is the diameter of the circle. We are also given the lengths of the other two sides: AB = 8 and BC = 15.
To solve for the length of AC, we need to use the fact that when a triangle is inscribed in a circle, the angle formed by the diameter and the other side of the triangle is always a right angle (90 degrees).
We can use the Pythagorean theorem to find the missing side and calculate the length of the diameter (AC).
Let's label the intersection point of the diameter BC and the side AB as point P. We can then split the triangle into two right-angled triangles: ΔAPB and ΔCAB.
In triangle ΔAPB, we have:
AB = 8 (given)
AP = x (unknown)
Using the Pythagorean theorem, we can write:
AB² + AP² = BP²
In triangle ΔCAB, we have:
BC = 15 (given)
AC = x (unknown)
Using the Pythagorean theorem, we can write:
AB² + BC² = AC²
We can equate AP² and BP² since they both represent the same length squared:
AB² + AP² = BP² (equation 1)
Substituting the given values:
8² + x² = BP²
Since BP is the diameter BC, which equals 15, we can write:
8² + x² = 15²
64 + x² = 225
Subtracting 64 from both sides:
x² = 225 - 64
x² = 161
Taking the square root of both sides to solve for x:
x = √161
Therefore, the length of AP is √161.
To find the length of AC (the diameter), we can substitute the value of √161 into equation 2:
AB² + BC² = AC²
Substituting the given values:
8² + 15² = AC²
64 + 225 = AC²
289 = AC²
Taking the square root of both sides to solve for AC:
AC = √289
AC = 17
Thus, the length of the diameter AC is 17 units.