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13 problems match “Pythagorean theorem”. Clear

Longest side of a triangle ●○○○○ 10 ptsA triangle has vertices at coordinates $(2,2)$ , $(5,6)$ and $(6,2)$ . What is the length, in units, of the longest side of the triangle?
Third side of a 30-60-90 triangle ●○○○○ 10 ptsA right triangle has two interior angles of $30^\circ$ and $90^\circ$ , and two of its side lengths are $5$ and $5\sqrt{3}$ . Find the third side length.
Area of a square from vertices ●○○○○ 10 ptsWhat is the area, in square units, of the square with the four vertices at $A\ (0, 0)$ , $B\ (-5, -1)$ , $C\ (-4, -6)$ and $D\ (1, -5)$ ?
Altitude of an isosceles triangle ●○○○○ 10 ptsIn triangle $ABC$ with $AB=AC=25$ and $BC=14$ , let $D$ be the foot of the altitude from $A$ to $BC$ . Find the length of $AD$ .
Sum of legs in a 30-60-90 triangle ●○○○○ 10 ptsA 30-60-90 right triangle has hypotenuse $2\sqrt{6}$ centimeters. Find the sum of the lengths, in centimeters, of its two legs. Round to two decimal places.
Sine minus cosine from a terminal point ●○○○○ 10 ptsThe terminal side of angle $\alpha$ passes through the point $P(-2,4)$ . Find the value of $\sin\alpha - \cos\alpha$ . Round to two decimal places.
Area of circle through three points ●●○○○ 20 ptsA circle passes through the points $(1,1)$ , $(1,7)$ , and $(9,1)$ . The area of the circle can be expressed as $k\pi$ . Find the value of $k$ .
Minimum area with given hypotenuse altitude ●●○○○ 20 ptsIn right triangle $ABC$ , the altitude to the hypotenuse has length 4. Find the minimum value of the area of triangle $ABC$ .
Diagonal of a rectangular face ●●○○○ 20 ptsA regular right pentagonal prism has a length (the distance between its two pentagonal bases) of 18.9 cm and a volume of 452.7 cm $^3$ . Find the diagonal length of one of its rectangular faces, in cm, rounded to the nearest millimetre (one decimal place). Use the area of a regular pentagon $\approx 1.720477\,s^2$ for side length $s$ .
Difference of cone volumes from a triangle ●●○○○ 20 ptsA right-angled triangle with sides 6 cm, 8 cm and 10 cm is revolved about each of the two sides that contain the right angle, giving two solids. Find the difference between their volumes in cm $^3$ . Round to two decimal places.