Answer:
[tex]d=\dfrac{3 \sqrt{2}}{2}[/tex]
[tex]c=6\sqrt{3}[/tex]
Step-by-step explanation:
The first right triangle is a special 45-45-90 triangle because its interior angles measure 45°, 45° and 90°.
The side lengths of a 45-45-90 triangle are in the ratio 1 : 1 : √2, which means that the legs are congruent to each other, and the length of the hypotenuse is equal to the length of one leg multiplied by √2.
Given that the hypotenuse of this triangle is 3, then the length of the leg d can be calculated by dividing 3 by √2:
[tex]d=\dfrac{3}{\sqrt{2}} \\\\\\ d=\dfrac{3\cdot \sqrt{2}}{\sqrt{2}\cdot \sqrt{2}} \\\\\\ d=\dfrac{3 \sqrt{2}}{2}[/tex]
[tex]\dotfill[/tex]
The second right triangle is a special 30-60-90 triangle because its interior angles measure 30°, 60° and 90°.
The side lengths of a 30-60-90 triangle are in the ratio 1 : √3 : 2, which means that the length of the longest leg is √3 times the length of the shortest leg, and the hypotenuse is twice the length of the shortest leg.
Given that the length of the shortest leg (opposite the 30° angle) measures 6 units, the length of the longest leg c is √3 times this:
[tex]c=6 \cdot \sqrt{3} \\\\\\ c=6\sqrt{3}[/tex]
