Answer:
D. 36 units
Explanation:
You want the perimeter of the given isosceles triangle.
Pythagorean theorem
The Pythagorean theorem tells you the relationship between the side lengths of a right triangle. An altitude of an isosceles triangle divides it into two congruent right triangles. We can use this fact to find the lengths of the sides of the triangle that are not aligned with the grid.
By counting grid squares, or by finding the difference of coordinates, we can determine the lengths of the sides of each right triangle to be 12 units (horizontally) and 5 units (vertically). Then the slant length s is ...
s² = 12² + 5²
s² = 144 +25 = 169
s = √169 = 13
Perimeter
The two congruent side lengths of the red triangle are 13 units each, and its "base" is 10 units. The perimeter is the sum of these lengths:
P = 13 +13 +10 = 36 . . . . units
__
Additional comment
The integer side lengths of the right triangle can be written as a triple: {5, 12, 13}. This is known as a "Pythagorean triple." Perhaps the most famous of these is {3, 4, 5}. Others you will encounter in algebra, trig, and geometry problems are {7, 24, 25}, {8, 15, 17}.
It can be time-saving to learn to recognize such ratios of side lengths, so you don't have to go through the whole Pythagorean theorem calculation when you want to find a missing side.
You can also find the perimeter of the given triangle by considering the red sides of the right triangle: 5 + 13 = 18. They match corresponding lengths on the other half of the red triangle, so its perimeter is 2×18 = 36.